Definition df-ushgr 29039

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 29037	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1540	. . . . . . 7 class 𝑒
4	3	cdm 5619	. . . . . 6 class dom 𝑒
5		vv	. . . . . . . . 9 setvar 𝑣
6	5	cv 1540	. . . . . . . 8 class 𝑣
7	6	cpw 4549	. . . . . . 7 class 𝒫 𝑣
8		c0 4282	. . . . . . . 8 class ∅
9	8	csn 4575	. . . . . . 7 class {∅}
10	7, 9	cdif 3895	. . . . . 6 class (𝒫 𝑣 ∖ {∅})
11	4, 10, 3	wf1 6483	. . . . 5 wff 𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
12		vg	. . . . . . 7 setvar 𝑔
13	12	cv 1540	. . . . . 6 class 𝑔
14		ciedg 28977	. . . . . 6 class iEdg
15	13, 14	cfv 6486	. . . . 5 class (iEdg‘𝑔)
16	11, 2, 15	wsbc 3737	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
17		cvtx 28976	. . . . 5 class Vtx
18	13, 17	cfv 6486	. . . 4 class (Vtx‘𝑔)
19	16, 5, 18	wsbc 3737	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})
20	19, 12	cab 2711	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}
21	1, 20	wceq 1541	1 wff USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})}

This definition is referenced by:  isushgr  29041