Definition df-ushgrm 16294

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 Jim Kingdon, 31-Dec-2025.)

Assertion

Ref	Expression
df-ushgrm	⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}

Distinct variable group:   𝑒,𝑔,𝑗,𝑠,𝑣

Detailed syntax breakdown of Definition df-ushgrm

Step	Hyp	Ref	Expression
1		cushgr 16292	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1401	. . . . . . 7 class 𝑒
4	3	cdm 4772	. . . . . 6 class dom 𝑒
5		vj	. . . . . . . . 9 setvar 𝑗
6		vs	. . . . . . . . 9 setvar 𝑠
7	5, 6	wel 2210	. . . . . . . 8 wff 𝑗 ∈ 𝑠
8	7, 5	wex 1545	. . . . . . 7 wff ∃𝑗 𝑗 ∈ 𝑠
9		vv	. . . . . . . . 9 setvar 𝑣
10	9	cv 1401	. . . . . . . 8 class 𝑣
11	10	cpw 3688	. . . . . . 7 class 𝒫 𝑣
12	8, 6, 11	crab 2532	. . . . . 6 class {𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
13	4, 12, 3	wf1 5372	. . . . 5 wff 𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
14		vg	. . . . . . 7 setvar 𝑔
15	14	cv 1401	. . . . . 6 class 𝑔
16		ciedg 16237	. . . . . 6 class iEdg
17	15, 16	cfv 5375	. . . . 5 class (iEdg‘𝑔)
18	13, 2, 17	wsbc 3051	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
19		cvtx 16236	. . . . 5 class Vtx
20	15, 19	cfv 5375	. . . 4 class (Vtx‘𝑔)
21	18, 9, 20	wsbc 3051	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
22	21, 14	cab 2224	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}
23	1, 22	wceq 1402	1 wff USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}

This definition is referenced by:  isushgrm  16296