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Definition df-ushgr 26153
Description: 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.)
Assertion
Ref Expression
df-ushgr USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
Distinct variable group:   𝑒,𝑔,𝑣

Detailed syntax breakdown of Definition df-ushgr
StepHypRef Expression
1 cushgr 26151 . 2 class USHGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1631 . . . . . . 7 class 𝑒
43cdm 5266 . . . . . 6 class dom 𝑒
5 vv . . . . . . . . 9 setvar 𝑣
65cv 1631 . . . . . . . 8 class 𝑣
76cpw 4302 . . . . . . 7 class 𝒫 𝑣
8 c0 4058 . . . . . . . 8 class
98csn 4321 . . . . . . 7 class {∅}
107, 9cdif 3712 . . . . . 6 class (𝒫 𝑣 ∖ {∅})
114, 10, 3wf1 6046 . . . . 5 wff 𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
12 vg . . . . . . 7 setvar 𝑔
1312cv 1631 . . . . . 6 class 𝑔
14 ciedg 26074 . . . . . 6 class iEdg
1513, 14cfv 6049 . . . . 5 class (iEdg‘𝑔)
1611, 2, 15wsbc 3576 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
17 cvtx 26073 . . . . 5 class Vtx
1813, 17cfv 6049 . . . 4 class (Vtx‘𝑔)
1916, 5, 18wsbc 3576 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
2019, 12cab 2746 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
211, 20wceq 1632 1 wff USHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
Colors of variables: wff setvar class
This definition is referenced by:  isushgr  26155
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