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Definition df-umgr 26023
Description: Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This 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." To provide uniform definitions for all kinds of graphs, 𝑥 ∈ (𝒫 𝑣 ∖ {∅}) is used as restriction of the class abstraction, although 𝑥 ∈ 𝒫 𝑣 would be sufficient (see prprrab 13293 and isumgrs 26036). (Contributed by AV, 24-Nov-2020.)
Assertion
Ref Expression
df-umgr UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-umgr
StepHypRef Expression
1 cumgr 26021 . 2 class UMGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1522 . . . . . . 7 class 𝑒
43cdm 5143 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1522 . . . . . . . . 9 class 𝑥
7 chash 13157 . . . . . . . . 9 class #
86, 7cfv 5926 . . . . . . . 8 class (#‘𝑥)
9 c2 11108 . . . . . . . 8 class 2
108, 9wceq 1523 . . . . . . 7 wff (#‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1522 . . . . . . . . 9 class 𝑣
1312cpw 4191 . . . . . . . 8 class 𝒫 𝑣
14 c0 3948 . . . . . . . . 9 class
1514csn 4210 . . . . . . . 8 class {∅}
1613, 15cdif 3604 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 2945 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
184, 17, 3wf 5922 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1522 . . . . . 6 class 𝑔
21 ciedg 25920 . . . . . 6 class iEdg
2220, 21cfv 5926 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3468 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
24 cvtx 25919 . . . . 5 class Vtx
2520, 24cfv 5926 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3468 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
2726, 19cab 2637 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
281, 27wceq 1523 1 wff UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
Colors of variables: wff setvar class
This definition is referenced by:  isumgr  26035
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