Definition df-ushgrm 15924

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 15922	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar e
3	2	cv 1396	. . . . . . 7 class e
4	3	cdm 4725	. . . . . 6 class dom e
5		vj	. . . . . . . . 9 setvar j
6		vs	. . . . . . . . 9 setvar s
7	5, 6	wel 2203	. . . . . . . 8 wff j e. s
8	7, 5	wex 1540	. . . . . . 7 wff E. j j e. s
9		vv	. . . . . . . . 9 setvar v
10	9	cv 1396	. . . . . . . 8 class v
11	10	cpw 3652	. . . . . . 7 class ~P v
12	8, 6, 11	crab 2514	. . . . . 6 class { s e. ~P v | E. j j e. s }
13	4, 12, 3	wf1 5323	. . . . 5 wff e : dom e -1-1-> { s e. ~P v | E. j j e. s }
14		vg	. . . . . . 7 setvar g
15	14	cv 1396	. . . . . 6 class g
16		ciedg 15867	. . . . . 6 class iEdg
17	15, 16	cfv 5326	. . . . 5 class (iEdg ` g )
18	13, 2, 17	wsbc 3031	. . . 4 wff [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s }
19		cvtx 15866	. . . . 5 class Vtx
20	15, 19	cfv 5326	. . . 4 class (Vtx ` g )
21	18, 9, 20	wsbc 3031	. . 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 2217	. 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 1397	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  15926