Definition df-ushgr 29144

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 29142	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1541	. . . . . . 7 class 𝑒
4	3	cdm 5632	. . . . . 6 class dom 𝑒
5		vv	. . . . . . . . 9 setvar 𝑣
6	5	cv 1541	. . . . . . . 8 class 𝑣
7	6	cpw 4556	. . . . . . 7 class 𝒫 𝑣
8		c0 4287	. . . . . . . 8 class ∅
9	8	csn 4582	. . . . . . 7 class {∅}
10	7, 9	cdif 3900	. . . . . 6 class (𝒫 𝑣 ∖ {∅})
11	4, 10, 3	wf1 6497	. . . . 5 wff 𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
12		vg	. . . . . . 7 setvar 𝑔
13	12	cv 1541	. . . . . 6 class 𝑔
14		ciedg 29082	. . . . . 6 class iEdg
15	13, 14	cfv 6500	. . . . 5 class (iEdg‘𝑔)
16	11, 2, 15	wsbc 3742	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
17		cvtx 29081	. . . . 5 class Vtx
18	13, 17	cfv 6500	. . . 4 class (Vtx‘𝑔)
19	16, 5, 18	wsbc 3742	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
20	19, 12	cab 2715	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
21	1, 20	wceq 1542	1 wff USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}

This definition is referenced by:  isushgr  29146