Definition df-ushgr 29619

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 29617	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1569	. . . . . . 7 class 𝑒
4	3	cdm 5651	. . . . . 6 class dom 𝑒
5		vv	. . . . . . . . 9 setvar 𝑣
6	5	cv 1569	. . . . . . . 8 class 𝑣
7	6	cpw 4557	. . . . . . 7 class 𝒫 𝑣
8		c0 4279	. . . . . . . 8 class ∅
9	8	csn 4584	. . . . . . 7 class {∅}
10	7, 9	cdif 3896	. . . . . 6 class (𝒫 𝑣 ∖ {∅})
11	4, 10, 3	wf1 6528	. . . . 5 wff 𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
12		vg	. . . . . . 7 setvar 𝑔
13	12	cv 1569	. . . . . 6 class 𝑔
14		ciedg 29557	. . . . . 6 class iEdg
15	13, 14	cfv 6531	. . . . 5 class (iEdg‘𝑔)
16	11, 2, 15	wsbc 3739	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
17		cvtx 29556	. . . . 5 class Vtx
18	13, 17	cfv 6531	. . . 4 class (Vtx‘𝑔)
19	16, 5, 18	wsbc 3739	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
20	19, 12	cab 2739	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
21	1, 20	wceq 1570	1 wff USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}

This definition is used by:  isushgr  29621