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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxcont 25901* The axiom of continuity. Take two sets of points 𝐴 and 𝐵. If all the points in 𝐴 come before the points of 𝐵 on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)
((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥𝐴𝑦𝐵 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥𝐴𝑦𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

15.4.2.3  EE^n fulfills Tarski's Axioms

Syntaxceeng 25902 Extends class notation with the Tarski geometry structure for 𝔼↑𝑁.
class EEG

Definitiondf-eeng 25903* Define the geometry structure for 𝔼↑𝑁. (Contributed by Thierry Arnoux, 24-Aug-2017.)
EEG = (𝑛 ∈ ℕ ↦ ({⟨(Base‘ndx), (𝔼‘𝑛)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥𝑖) − (𝑦𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ ((𝔼‘𝑛) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theoremeengv 25904* The value of the Euclidean geometry for dimension 𝑁. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) = ({⟨(Base‘ndx), (𝔼‘𝑁)⟩, ⟨(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥𝑖) − (𝑦𝑖))↑2))⟩} ∪ {⟨(Itv‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ 𝑧 Btwn ⟨𝑥, 𝑦⟩})⟩, ⟨(LineG‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn ⟨𝑥, 𝑦⟩ ∨ 𝑥 Btwn ⟨𝑧, 𝑦⟩ ∨ 𝑦 Btwn ⟨𝑥, 𝑧⟩)})⟩}))

Theoremeengstr 25905 The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) Struct ⟨1, 17⟩)

Theoremeengbas 25906 The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))

Theoremebtwntg 25907 The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝜑𝑁 ∈ ℕ)    &   𝑃 = (Base‘(EEG‘𝑁))    &   𝐼 = (Itv‘(EEG‘𝑁))    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)       (𝜑 → (𝑍 Btwn ⟨𝑋, 𝑌⟩ ↔ 𝑍 ∈ (𝑋𝐼𝑌)))

Theoremecgrtg 25908 The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝜑𝑁 ∈ ℕ)    &   𝑃 = (Base‘(EEG‘𝑁))    &    = (dist‘(EEG‘𝑁))    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)       (𝜑 → (⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩ ↔ (𝐴 𝐵) = (𝐶 𝐷)))

Theoremelntg 25909* The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)
𝑃 = (Base‘(EEG‘𝑁))    &   𝐼 = (Itv‘(EEG‘𝑁))       (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑧𝑃 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}))

Theoremeengtrkg 25910 The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Theoremeengtrkge 25911 The geometry structure for 𝔼↑𝑁 is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)
(𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiGE)

PART 16  GRAPH THEORY

To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to definitions in [Bollobas] p. 1-8 or in [Diestel] p. 2-28. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

# Basic concepts:

TermReferenceDefinitionRemarks
Vertex df-vtx 25921 A vertex of a graph 𝐺 is an element of the set of vertices (Vtx‘𝐺) of the graph 𝐺. The set of vertices (Vtx‘𝐺) (corresponding to V(G) in [Bollobas] p. 1) is usually the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 (see opvtxfv 25929), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 25946).
Edge df-edg 25985 An edge of a graph 𝐺 is a nonempty set of vertices of the graph. It is said that these vertices are "joined" or "connected" by the edge, see [Bollobas] p. 1. The set of edges (Edg‘𝐺) (corresponding to E(G) in [Bollobas] p. 1) is usually the range ran 𝐸 of the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸, or the range of the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure.
Loop A loop in a graph 𝐺 is an edge which connects a single vertex with itself (or, according to [Bollobas] p. 7 "joins a vertex to itself"). In other words, a loop is an edge 𝑒 ∈ (Edg‘𝐺) which is a singleton consisting of a vertex 𝑣 ∈ (Vtx‘𝐺): 𝑒 = {𝑣}
Edge function resp. indexed edges df-iedg 25922 An edge function (or indexed set of edges) of a graph 𝐺 is a mapping of an arbitrary index set to nonempty sets of vertices of the graph. The edge function (iEdg‘𝐺) is usually the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 > (see opiedgfv 25932), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 25947).
The set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 25986.
Whereas the concept of plain edges is sufficient for simple hypergraphs, indexed edges are required for e.g. multigraphs in which the same vertices may be connected by more than one edge.

# Basic kinds of graphs:

TermReferenceDefinitionRemarks
Undirected hypergraph df-uhgr 25998 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}). In this most general definition of a graph, an "edge" may connect three or more vertices with each other, see [Berge] p. 1.
In Wikipedia "Hypergraph", see https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020) such a hypergraph is called a "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure (18-Jan-2020) "Each hypergraph [...] can be regarded as an incidence structure in which the [vertices] play the role of "points", the corresponding family of [edges] plays the role of "lines" and the incidence relation is set membership".

Notice that by using (Edg‘𝐺) the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this representation will only be used for undirected simple hypergraphs.
Undirected simple hypergraph df-ushgr 25999 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the power set of the vertices 𝑉 = (Vtx‘𝐺): ran 𝐸 ⊆ (𝒫 𝑉 ∖ {∅}). See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020). This is how a "hypergraph" is defined in Section I.1 in [Bollobas] p. 7 or the definition in section 1.10 in [Diestel] p. 27. A simple hypergraph has at most one edge between the same vertices, hence a pseudograph needs not be a simple hypergraph.
According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E_1, E_2, ..., E_m } such that E_i C_ E_j => i = j". By this definition, a simple hypergraph cannot contain the edges E_1 = { v_1 , v_2 } and E_2 = { v_1, v_2, v_3 }, because E_1 C_ E_2, but 1 =/= 2.
Undirected loop-free hypergraph--- an undirected hypergraph without a loop, i.e. all edges connect at least two vertices.
Undirected pseudograph df-upgr 26022 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 4424). This means a pseudograph may contain loops.
This defintion corresponds to the definition of a "multigraph" in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed", or in [Diestel] p. 28, "A multigraph is a pair (V,E) of disjoint sets (of vertices and edges) together with a map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end(s).".
Undirected multigraph df-umgr 26023 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). This definition is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges."
A proper unordered pair contains two different elements, therefore a multigraph does not have loops.
Undirected simple pseudograph df-uspgr 26090 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper or not proper) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph df-usgr 26091 a class 𝐺 with an edge function 𝐸 = (iEdg‘𝐺) which is a one-to-one function into the set of (proper!) unordered pairs of vertices 𝑉 = (Vtx‘𝐺). An ordered pair 𝑉, 𝐸 of two distinct sets 𝑉 (the vertices) and 𝐸 (the edges), the "usual" definition of a "graph", see, for example, the definition in section I.1 of [Bollobas] p. 1 or in section 1.1 of [Diestel] p. 2, can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see usgrausgrb 26109.
Finite graph df-fusgr 26254 a graph 𝐺 with a finite set of vertices 𝑉 = (Vtx‘𝐺). See definitions in [Bollobas] p. 1 or [Diestel] p. 2.
In simple graphs, the set of (indexed) edges (iEdg‘𝐺) (and therefore also the set of edges (Edg‘𝐺)) is finite if 𝑉 = (Vtx‘𝐺) is finite, see fusgrfis 26267. The number of edges is limited by (𝑛C2) (or "𝑛 choose 2") with 𝑛 = (#‘𝑉), see fusgrmaxsize 26416. Analogously, the number of edges 𝐸 = (iEdg‘𝐺) of an undirected simple pseudograph (which may have loops) is limited by ((𝑛 + 1)C2). In pseudographs or multigraphs, however, 𝐸 can be infinite although 𝑉 is finite.
Graph of finite size--- a graph 𝐺 with a finite set 𝐸 = (iEdg‘𝐺), i.e. with a finite number of edges. A graph can be of finite size although its set of vertices is infinite (most of the vertices would not be connected by an edge).

# Terms and properties of graphs:

TermReferenceDefinitionRemarks
Edge joining resp. connecting (two) vertices --- An edge 𝑒 ∈ (Edg‘𝐺) joins resp. connects the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) if 𝑒 = { v_1, v_2, ... v_n }. If 𝑛 = 1, 𝑒 = { v_1 } is a loop, if 𝑛 = 2, 𝑒 = { v_1 , v_2 } is an edge as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge see definition in Section I.1 in [Bollobas] p. 1. If an edge 𝑒 ∈ (Edg‘𝐺) joins the vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ), then the vertices v_1, v_2, ... v_n are called the endvertices of the edge 𝑒.
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v_1, v_2, ... v_n (𝑛 ∈ ℕ) are adjacent if there is an edge e = { v_1, v_2, ... v_n } joining these vertices. In this case, the vertices are incident with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or connected by the edge e.
Edge ending at a vertex An edge 𝑒 ∈ (Edg‘𝐺) is ending at a vertex 𝑣 if the vertex is an endvertex of the edge: 𝑣𝑒. In other words, the vertex 𝑣 is incident with the edge 𝑒.
(Two) Adjacent edges The edges e_0, e_1, ... e_n (𝑛 ∈ ℕ) are adjacent if they have exactly one common endvertex. Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph see definition in Section I.1 in [Bollobas] p. 3 The order of a graph 𝐺 is the number of vertices in the graph: (#‘(Vtx‘𝐺)).
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 The size of a graph 𝐺 is the number of edges in the graph: (#‘(iEdg‘𝐺)). Or, for a simple graph 𝐺: (#‘(Edg‘𝐺))).
Neighborhood of a vertex df-nbgr 26270 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex 𝑣 by an edge is called a neighbor of the vertex 𝑣. The set of neighbors of a vertex 𝑣 is called the neighborhood (or open neighborhood) of the vertex 𝑣. The closed neighborhood is the union of the (open) neighborhood of the vertex 𝑣 with {𝑣}.
Degree of a vertex df-vtxdg 26418 The degree of a vertex is the number of the edges ending at this vertex. In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex usgrvd0nedg 26485 A vertex is called isolated if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 26332 A vertex is called universal if it is connected with every other vertex of the graph by an edge, thus having degree ((#‘(Vtx‘𝐺)) − ).

# Special kinds of graphs:

TermReferenceDefinitionRemarks
Complete graph df-cplgr 26362 A graph is called complete if each pair of vertices is connected by an edge. The size of a complete undirected simple graph of order 𝑛 is (𝑛C2) (or "𝑛 choose 2"), see cusgrsize 26406.
Empty graph uhgr0e 26011 A graph is called empty if it has no edges.
Null graph uhgr0 26013 and uhgr0vb 26012 A graph is called a null graph if it has no vertices (and therefore also no edges).
Trivial graph usgr1v 26193 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngr 27165 resp. definition in Section I.1 in [Bollobas] p. 6 A graph is called connected if for each pair of vertices there is a path between these vertices.

For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.

16.1  Vertices and edges

In the following, the vertices and (indexed) edges for an arbitrary class 𝐺 (called "graph" in the following) are defined and examined. The main result of this section is to show that the set of vertices (Vtx‘𝐺) of a graph 𝐺 is the first component 𝑉 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 (see opvtxfv 25929), or the base set (Base‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see basvtxval 25946), and that the set of indexed edges resp. the edge function (iEdg‘𝐺) is the second component 𝐸 of the graph 𝐺 if it is represented by an ordered pair 𝑉, 𝐸 (see opiedgfv 25932), or the component (.ef‘𝐺) of the graph 𝐺 if it is represented as extensible structure (see edgfiedgval 25947). Finally, it is shown that the set of edges of a graph 𝐺 is the range of its edge function: (Edg‘𝐺) = ran (iEdg‘𝐺), see edgval 25986.

Usually, a graph 𝐺 is a set. If 𝐺 is a proper class, however, it represents the null graph (without vertices and edges), because (Vtx‘𝐺) = ∅ and (iEdg‘𝐺) = ∅ holds, see vtxvalprc 25982 and iedgvalprc 25983.

Up to the end of this section, the edges need not be related to the vertices. Once undirected hypergraphs are defined (see df-uhgr 25998), the edges become nonempty sets of vertices, and by this obtain their meaning as "connectors" of vertices.

16.1.1  The edge function extractor for extensible structures

Syntaxcedgf 25912 Extend class notation with an edge function.
class .ef

Definitiondf-edgf 25913 Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)
.ef = Slot 18

Theoremedgfid 25914 Utility theorem: index-independent form of df-edgf 25913. (Contributed by AV, 16-Nov-2021.)
.ef = Slot (.ef‘ndx)

Theoremedgfndxnn 25915 The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.)
(.ef‘ndx) ∈ ℕ

Theoremedgfndxid 25916 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)
(𝐺𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))

Theorembaseltedgf 25917 The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) < (.ef‘ndx)

Theoremslotsbaseefdif 25918 The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)
(Base‘ndx) ≠ (.ef‘ndx)

16.1.2  Vertices and indexed edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/pseudo-/ multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath: 𝑉, 𝐸.

Another way is to regard a graph as a mathematical structure, which consistes at least of a set (of vertices) and a relation between the vertices (edge function), but which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 15906.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 25921 and df-iedg 25922), which provide the vertices resp. (indexed) edges of an arbitrary class 𝐺 which represents a graph: (Vtx‘𝐺) resp. (iEdg‘𝐺). In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G).").

Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. For example, e1 = e(1) = { a, b } and e2 = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e1 and e2 connecting the same two vertices a and b, we would have e(e1) = e(e2) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair (𝐺 ∈ (V × V)), the set of vertices is (Vtx‘𝐺) = (1st𝐺) (see opvtxval 25928) and the set of (indexed) edges is (iEdg‘𝐺) = (2nd𝐺) (see opiedgval 25931), or if 𝐺 is given as ordered pair 𝐺 = ⟨𝑉, 𝐸, the set of vertices is (Vtx‘𝐺) = 𝑉 (see opvtxfv 25929) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see opiedgfv 25932).

And for a graph represented as extensible structure (𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩), the set of vertices is (Vtx‘𝐺) = (Base‘𝐺) (see funvtxval 25950) and the set of (indexed) edges is (iEdg‘𝐺) = (.ef‘𝐺) (see funiedgval 25951), or if 𝐺 is given in its simplest form as extensible structure with two slots (𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), the set of vertices is (Vtx‘𝐺) = 𝑉 (see struct2grvtx 25964) and the set of (indexed) edges is (iEdg‘𝐺) = 𝐸 (see struct2griedg 25965).

These two representations are convertible, see graop 25966 and grastruct 25967: If 𝐺 is a graph (for example 𝐺 = ⟨𝑉, 𝐸), then 𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩} represents essentially the same graph, and if 𝐺 is a graph (for example 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}), then 𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ represents essentially the same graph. In both cases, (Vtx‘𝐺) = (Vtx‘𝐻) and (iEdg‘𝐺) = (iEdg‘𝐻) hold. Theorems gropd 25968 and gropeld 25970 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. Analogously, theorems grstructd 25969 and grstructeld 25971 show that if any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property, then any extensible structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is also such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be 𝐺 = ⟨𝑉, ∅⟩ or 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), ∅⟩}, a structure without a slot for edges can be used: 𝐺 = {⟨(Base‘ndx), 𝑉⟩}, see snstrvtxval 25974 and snstriedgval 25975. Analogously, the empty set can be used to represent the null graph, see vtxval0 25976 and iedgval0 25977, which can also be represented by 𝐺 = ⟨∅, ∅⟩ or 𝐺 = {⟨(Base‘ndx), ∅⟩, ⟨(.ef‘ndx), ∅⟩}. Even proper classes can be used to represent the null graph, see vtxvalprc 25982 and iedgvalprc 25983.

Other classes should not be used to represent graphs, because there could be a degenerated behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 25978 resp. iedgvalsnop 25979, and vtxval3sn 25980 resp. iedgval3sn 25981.

16.1.2.1  Definitions and basic properties

Syntaxcvtx 25919 Extend class notation with the vertices of "graphs".
class Vtx

Syntaxciedg 25920 Extend class notation with the indexed edges of "graphs".
class iEdg

Definitiondf-vtx 25921 Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)
Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))

Definitiondf-iedg 25922 Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)
iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)))

Theoremvtxval 25923 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
(Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))

Theoremiedgval 25924 The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
(iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺))

TheoremvtxvalOLD 25925 Obsolete version of vtxval 25923 as of 11-Nov-2021. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))

TheoremiedgvalOLD 25926 Obsolete version of iedgval 25924 as of 11-Nov-2021. (Contributed by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺𝑉 → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))

Theorem1vgrex 25927 A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉𝐺 ∈ V)

16.1.2.2  The vertices and edges of a graph represented as ordered pair

Theoremopvtxval 25928 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
(𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st𝐺))

Theoremopvtxfv 25929 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Theoremopvtxov 25930 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (𝑉Vtx𝐸) = 𝑉)

Theoremopiedgval 25931 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
(𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd𝐺))

Theoremopiedgfv 25932 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Theoremopiedgov 25933 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
((𝑉𝑋𝐸𝑌) → (𝑉iEdg𝐸) = 𝐸)

Theoremopvtxfvi 25934 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
𝑉 ∈ V    &   𝐸 ∈ V       (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Theoremopiedgfvi 25935 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
𝑉 ∈ V    &   𝐸 ∈ V       (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸

16.1.2.3  The vertices and edges of a graph represented as extensible structure

Theoremfunvtxdmge2val 25936 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdmge2val 25937 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremfunvtxdm2val 25938 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun (𝐺 ∖ {∅}) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdm2val 25939 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun (𝐺 ∖ {∅}) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremfunvtxdm2valOLD 25940 Obsolete version of funvtxdm2val 25938 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdm2valOLD 25941 Obsolete version of funiedgdm2val 25939 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐵 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremfunvtxval0 25942 The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
𝑆 ∈ V       ((Fun (𝐺 ∖ {∅}) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfunvtxval0OLD 25943 Obsolete version of funvtxval0 25942 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑆 ∈ V       (((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅})) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfunvtxdmge2valOLD 25944 Obsolete version of funvtxdmge2val 25936 as of 11-Nov-2021. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgdmge2valOLD 25945 Obsolete version of funiedgdmge2val 25937 as of 11-Nov-2021. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺))

Theorembasvtxval 25946 The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

Theoremedgfiedgval 25947 The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝐸𝑌)    &   (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

TheorembasvtxvalOLD 25948 Obsolete version of basvtxval 25946 as of 12-Nov-2021. (Contributed by AV, 14-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝑉𝑌)    &   (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

TheoremedgfiedgvalOLD 25949 Obsolete version of edgfiedgval 25947 as of 12-Nov-2021. (Contributed by AV, 14-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 2 ≤ (#‘dom 𝐺))    &   (𝜑𝐸𝑌)    &   (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

Theoremfunvtxval 25950 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

Theoremfuniedgval 25951 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

TheoremfunvtxvalOLD 25952 Obsolete version of funvtxval 25950 as of 12-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺))

TheoremfuniedgvalOLD 25953 Obsolete version of funiedgval 25951 as of 12-Nov-2021. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐺𝑉 ∧ Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺))

Theoremstructvtxvallem 25954 Lemma for structvtxval 25955 and structiedg0val 25956. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → 2 ≤ (#‘dom 𝐺))

Theoremstructvtxval 25955 The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)

Theoremstructiedg0val 25956 The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝑆 ∈ ℕ    &   (Base‘ndx) < 𝑆    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩}       ((𝑉𝑋𝐸𝑌𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅)

Theoremstructgrssvtxlem 25957 Lemma for structgrssvtx 25958 and structgrssiedg 25959. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → 2 ≤ (#‘dom 𝐺))

Theoremstructgrssvtx 25958 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

Theoremstructgrssiedg 25959 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
(𝜑𝐺 Struct 𝑋)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

TheoremstructgrssvtxlemOLD 25960 Obsolete version of structgrssvtxlem 25957 as of 14-Nov-2021. (Contributed by AV, 14-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → 2 ≤ (#‘dom 𝐺))

TheoremstructgrssvtxOLD 25961 Obsolete version of structgrssvtx 25958 as of 14-Nov-2021. (Contributed by AV, 14-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (Vtx‘𝐺) = 𝑉)

TheoremstructgrssiedgOLD 25962 Obsolete version of structgrssiedg 25959 as of 14-Nov-2021. (Contributed by AV, 14-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐺𝑋)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝑉𝑌)    &   (𝜑𝐸𝑍)    &   (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺)       (𝜑 → (iEdg‘𝐺) = 𝐸)

Theoremstruct2grstr 25963 A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       𝐺 Struct ⟨(Base‘ndx), (.ef‘ndx)⟩

Theoremstruct2grvtx 25964 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (Vtx‘𝐺) = 𝑉)

Theoremstruct2griedg 25965 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑋𝐸𝑌) → (iEdg‘𝐺) = 𝐸)

Theoremgraop 25966 Any representation of a graph 𝐺 (especially as extensible structure 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)
𝐻 = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Theoremgrastruct 25967 Any representation of a graph 𝐺 (especially as ordered pair 𝐺 = ⟨𝑉, 𝐸) is convertible in a representation of the graph as extensible structure. (Contributed by AV, 8-Oct-2020.)
𝐻 = {⟨(Base‘ndx), (Vtx‘𝐺)⟩, ⟨(.ef‘ndx), (iEdg‘𝐺)⟩}       ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))

Theoremgropd 25968* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 11-Oct-2020.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)

Theoremgrstructd 25969* If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (#‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑[𝑆 / 𝑔]𝜓)

Theoremgropeld 25970* If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 11-Oct-2020.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)       (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)

Theoremgrstructeld 25971* If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
(𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))    &   (𝜑𝑉𝑈)    &   (𝜑𝐸𝑊)    &   (𝜑𝑆𝑋)    &   (𝜑 → Fun (𝑆 ∖ {∅}))    &   (𝜑 → 2 ≤ (#‘dom 𝑆))    &   (𝜑 → (Base‘𝑆) = 𝑉)    &   (𝜑 → (.ef‘𝑆) = 𝐸)       (𝜑𝑆𝐶)

Theoremsetsvtx 25972 The vertices of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 18-Jan-2020.) (Revised by AV, 16-Nov-2021.)
𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct 𝑋)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐸𝑊)       (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Base‘𝐺))

Theoremsetsiedg 25973 The (indexed) edges of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct 𝑋)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐸𝑊)       (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸)

16.1.2.4  Representations of graphs without edges

Theoremsnstrvtxval 25974 The set of vertices of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See vtxvalsnop 25978 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 23-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (Vtx‘𝐺) = 𝑉)

Theoremsnstriedgval 25975 The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 25979 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 24-Sep-2020.)
𝑉 ∈ V    &   𝐺 = {⟨(Base‘ndx), 𝑉⟩}       (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅)

16.1.2.5  Degenerated cases of representations of graphs

Theoremvtxval0 25976 Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(Vtx‘∅) = ∅

Theoremiedgval0 25977 Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
(iEdg‘∅) = ∅

Theoremvtxvalsnop 25978 Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (Vtx‘𝐺) = {𝐵}

Theoremiedgvalsnop 25979 Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
𝐵 ∈ V    &   𝐺 = {⟨𝐵, 𝐵⟩}       (iEdg‘𝐺) = {𝐵}

Theoremvtxval3sn 25980 Degenerated case 3 for vertices: The set of vertices of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V       (Vtx‘{{{𝐴}}}) = {𝐴}

Theoremiedgval3sn 25981 Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.)
𝐴 ∈ V       (iEdg‘{{{𝐴}}}) = {𝐴}

Theoremvtxvalprc 25982 Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
(𝐶 ∉ V → (Vtx‘𝐶) = ∅)

Theoremiedgvalprc 25983 Degenerated case 4 for edges: The set of indexed edges of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
(𝐶 ∉ V → (iEdg‘𝐶) = ∅)

16.1.3  Edges as range of the edge function

Syntaxcedg 25984 Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).
class Edg

Definitiondf-edg 25985 Define the class of edges of a graph, see also definition "E = E(G)" in section I.1 of [Bollobas] p. 1. This definition is very general: It defines edges of a class as the range of its edge function (which does not even need to be a function). Therefore, this definition could also be used for hypergraphs, pseudographs and multigraphs. In these cases, however, the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. In some cases, this is no problem, so theorems with Edg are meaningful nevertheless (e.g., edguhgr 26069). Usually, however, this definition is used only for undirected simple (hyper-/pseudo-)graphs (with or without loops). (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))

Theoremedgval 25986 The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.)
(Edg‘𝐺) = ran (iEdg‘𝐺)

TheoremedgvalOLD 25987 Obsolete version of edgval 25986 as of 8-Dec-2021. (Contributed by AV, 1-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Theoremiedgedg 25988 An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.)
𝐸 = (iEdg‘𝐺)       ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ (Edg‘𝐺))

Theoremedgopval 25989 The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
((𝑉𝑊𝐸𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Theoremedgov 25990 The edges of a graph represented as ordered pair, shown as operation value. Although a little less intuitive, this representation is often used because it is shorter than the representation as function value of a graph given as ordered pair, see edgopval 25989. The representation ran 𝐸 for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 13-Oct-2020.)
((𝑉𝑊𝐸𝑋) → (𝑉Edg𝐸) = ran 𝐸)

Theoremedgstruct 25991 The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.)
𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝑉𝑊𝐸𝑋) → (Edg‘𝐺) = ran 𝐸)

Theoremedgiedgb 25992* A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.)
𝐼 = (iEdg‘𝐺)       (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))

TheoremedgiedgbOLD 25993* Obsolete version of edgiedgb 25992 as of 8-Dec-2021. (Contributed by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))

Theoremedg0iedg0 25994 There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅))

Theoremedg0iedg0OLD 25995 Obsolete version of edg0iedg0 25994 as of 8-Dec-2021. (Contributed by AV, 15-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺𝑊 ∧ Fun 𝐼) → (𝐸 = ∅ ↔ 𝐼 = ∅))

16.2  Undirected graphs

For undirected graphs, we will have the following hierarchy/taxonomy:

* Undirected Hypergraph: UHGraph

* Undirected loop-free graphs: ULFGraph (not defined formally yet)

* Undirected simple Hypergraph: USHGraph => USHGraph ⊆ UHGraph (ushgruhgr 26009)

* Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 26042)

* Undirected loop-free hypergraph: ULFHGraph (not defined formally yet) => ULFHGraph ⊆ UHGraph and ULFHGraph ULFGraph

* Undirected loop-free simple hypergraph: ULFSHGraph (not defined formally yet) => ULFSHGraph ⊆ USHGraph and ULFSHGraph ULFHGraph

* Undirected simple Pseudograph: USPGraph => USPGraph ⊆ UPGraph (uspgrupgr 26116) and USPGraph ⊆ USHGraph (uspgrushgr 26115), see also uspgrupgrushgr 26117

* Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 26043) and UMGraph ⊆ ULFHGraph (umgrislfupgr 26063)

* Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 26118) and USGraph ⊆ UMGraph (usgrumgr 26119) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 26124) see also usgrumgruspgr 26120

16.2.1  Undirected hypergraphs

Syntaxcuhgr 25996 Extend class notation with undirected hypergraphs.
class UHGraph

Syntaxcushgr 25997 Extend class notation with undirected simple hypergraphs.
class USHGraph

Definitiondf-uhgr 25998* Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into the power set of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 8-Oct-2020.)
UHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}

Definitiondf-ushgr 25999* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are non-empty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by AV, 8-Oct-2020.)
USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}

Theoremisuhgr 26000 The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

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