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Definition df-ushgr 26866
 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 nonempty 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 26864 . 2 class USHGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1537 . . . . . . 7 class 𝑒
43cdm 5520 . . . . . 6 class dom 𝑒
5 vv . . . . . . . . 9 setvar 𝑣
65cv 1537 . . . . . . . 8 class 𝑣
76cpw 4497 . . . . . . 7 class 𝒫 𝑣
8 c0 4243 . . . . . . . 8 class
98csn 4525 . . . . . . 7 class {∅}
107, 9cdif 3878 . . . . . 6 class (𝒫 𝑣 ∖ {∅})
114, 10, 3wf1 6324 . . . . 5 wff 𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
12 vg . . . . . . 7 setvar 𝑔
1312cv 1537 . . . . . 6 class 𝑔
14 ciedg 26804 . . . . . 6 class iEdg
1513, 14cfv 6327 . . . . 5 class (iEdg‘𝑔)
1611, 2, 15wsbc 3720 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
17 cvtx 26803 . . . . 5 class Vtx
1813, 17cfv 6327 . . . 4 class (Vtx‘𝑔)
1916, 5, 18wsbc 3720 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
2019, 12cab 2776 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
211, 20wceq 1538 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  26868
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