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Definition df-ushgr 26234
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 26232 . 2 class USHGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1651 . . . . . . 7 class 𝑒
43cdm 5279 . . . . . 6 class dom 𝑒
5 vv . . . . . . . . 9 setvar 𝑣
65cv 1651 . . . . . . . 8 class 𝑣
76cpw 4317 . . . . . . 7 class 𝒫 𝑣
8 c0 4081 . . . . . . . 8 class
98csn 4336 . . . . . . 7 class {∅}
107, 9cdif 3731 . . . . . 6 class (𝒫 𝑣 ∖ {∅})
114, 10, 3wf1 6067 . . . . 5 wff 𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
12 vg . . . . . . 7 setvar 𝑔
1312cv 1651 . . . . . 6 class 𝑔
14 ciedg 26169 . . . . . 6 class iEdg
1513, 14cfv 6070 . . . . 5 class (iEdg‘𝑔)
1611, 2, 15wsbc 3598 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
17 cvtx 26168 . . . . 5 class Vtx
1813, 17cfv 6070 . . . 4 class (Vtx‘𝑔)
1916, 5, 18wsbc 3598 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})
2019, 12cab 2751 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}
211, 20wceq 1652 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  26236
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