Definition df-ushgrm 15835

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 15833	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1374	. . . . . . 7 class 𝑒
4	3	cdm 4696	. . . . . 6 class dom 𝑒
5		vj	. . . . . . . . 9 setvar 𝑗
6		vs	. . . . . . . . 9 setvar 𝑠
7	5, 6	wel 2181	. . . . . . . 8 wff 𝑗 ∈ 𝑠
8	7, 5	wex 1518	. . . . . . 7 wff ∃𝑗 𝑗 ∈ 𝑠
9		vv	. . . . . . . . 9 setvar 𝑣
10	9	cv 1374	. . . . . . . 8 class 𝑣
11	10	cpw 3629	. . . . . . 7 class 𝒫 𝑣
12	8, 6, 11	crab 2492	. . . . . 6 class {𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
13	4, 12, 3	wf1 5291	. . . . 5 wff 𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
14		vg	. . . . . . 7 setvar 𝑔
15	14	cv 1374	. . . . . 6 class 𝑔
16		ciedg 15779	. . . . . 6 class iEdg
17	15, 16	cfv 5294	. . . . 5 class (iEdg‘𝑔)
18	13, 2, 17	wsbc 3008	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
19		cvtx 15778	. . . . 5 class Vtx
20	15, 19	cfv 5294	. . . 4 class (Vtx‘𝑔)
21	18, 9, 20	wsbc 3008	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}
22	21, 14	cab 2195	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑠 ∈ 𝒫 𝑣 ∣ ∃𝑗 𝑗 ∈ 𝑠}}
23	1, 22	wceq 1375	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  15837