Definition df-ushgrm 16052

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 Jim Kingdon, 31-Dec-2025.)

Assertion

Ref	Expression
df-ushgrm	|- USHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } }

Distinct variable group:   e, g, j, s, v

Detailed syntax breakdown of Definition df-ushgrm

Step	Hyp	Ref	Expression
1		cushgr 16050	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar e
3	2	cv 1397	. . . . . . 7 class e
4	3	cdm 4748	. . . . . 6 class dom e
5		vj	. . . . . . . . 9 setvar j
6		vs	. . . . . . . . 9 setvar s
7	5, 6	wel 2204	. . . . . . . 8 wff j e. s
8	7, 5	wex 1541	. . . . . . 7 wff E. j j e. s
9		vv	. . . . . . . . 9 setvar v
10	9	cv 1397	. . . . . . . 8 class v
11	10	cpw 3668	. . . . . . 7 class ~P v
12	8, 6, 11	crab 2524	. . . . . 6 class { s e. ~P v | E. j j e. s }
13	4, 12, 3	wf1 5348	. . . . 5 wff e : dom e -1-1-> { s e. ~P v | E. j j e. s }
14		vg	. . . . . . 7 setvar g
15	14	cv 1397	. . . . . 6 class g
16		ciedg 15995	. . . . . 6 class iEdg
17	15, 16	cfv 5351	. . . . 5 class (iEdg ` g )
18	13, 2, 17	wsbc 3041	. . . 4 wff [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s }
19		cvtx 15994	. . . . 5 class Vtx
20	15, 19	cfv 5351	. . . 4 class (Vtx ` g )
21	18, 9, 20	wsbc 3041	. . 3 wff [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s }
22	21, 14	cab 2218	. 2 class { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } }
23	1, 22	wceq 1398	1 wff USHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } }

This definition is referenced by:  isushgrm  16054