Definition df-ushgr 29091

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 29089	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1536	. . . . . . 7 class 𝑒
4	3	cdm 5689	. . . . . 6 class dom 𝑒
5		vv	. . . . . . . . 9 setvar 𝑣
6	5	cv 1536	. . . . . . . 8 class 𝑣
7	6	cpw 4605	. . . . . . 7 class 𝒫 𝑣
8		c0 4339	. . . . . . . 8 class ∅
9	8	csn 4631	. . . . . . 7 class {∅}
10	7, 9	cdif 3960	. . . . . 6 class (𝒫 𝑣 ∖ {∅})
11	4, 10, 3	wf1 6560	. . . . 5 wff 𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
12		vg	. . . . . . 7 setvar 𝑔
13	12	cv 1536	. . . . . 6 class 𝑔
14		ciedg 29029	. . . . . 6 class iEdg
15	13, 14	cfv 6563	. . . . 5 class (iEdg‘𝑔)
16	11, 2, 15	wsbc 3791	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
17		cvtx 29028	. . . . 5 class Vtx
18	13, 17	cfv 6563	. . . 4 class (Vtx‘𝑔)
19	16, 5, 18	wsbc 3791	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
20	19, 12	cab 2712	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
21	1, 20	wceq 1537	1 wff USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}

This definition is referenced by:  isushgr  29093