Definition df-ushgr 29318

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 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}

Distinct variable group:   𝑒,𝑔,𝑣

Detailed syntax breakdown of Definition df-ushgr

Step	Hyp	Ref	Expression
1		cushgr 29316	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1562	. . . . . . 7 class 𝑒
4	3	cdm 5652	. . . . . 6 class dom 𝑒
5		vv	. . . . . . . . 9 setvar 𝑣
6	5	cv 1562	. . . . . . . 8 class 𝑣
7	6	cpw 4558	. . . . . . 7 class 𝒫 𝑣
8		c0 4288	. . . . . . . 8 class ∅
9	8	csn 4585	. . . . . . 7 class {∅}
10	7, 9	cdif 3904	. . . . . 6 class (𝒫 𝑣 ∖ {∅})
11	4, 10, 3	wf1 6522	. . . . 5 wff 𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
12		vg	. . . . . . 7 setvar 𝑔
13	12	cv 1562	. . . . . 6 class 𝑔
14		ciedg 29256	. . . . . 6 class iEdg
15	13, 14	cfv 6525	. . . . 5 class (iEdg‘𝑔)
16	11, 2, 15	wsbc 3747	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
17		cvtx 29255	. . . . 5 class Vtx
18	13, 17	cfv 6525	. . . 4 class (Vtx‘𝑔)
19	16, 5, 18	wsbc 3747	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
20	19, 12	cab 2743	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
21	1, 20	wceq 1563	1 wff USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}

This definition is referenced by:  isushgr  29320