Definition df-ushgrm 15891

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 15889	. 2 class USHGraph
2		ve	. . . . . . . 8 setvar e
3	2	cv 1394	. . . . . . 7 class e
4	3	cdm 4720	. . . . . 6 class dom e
5		vj	. . . . . . . . 9 setvar j
6		vs	. . . . . . . . 9 setvar s
7	5, 6	wel 2201	. . . . . . . 8 wff j e. s
8	7, 5	wex 1538	. . . . . . 7 wff E. j j e. s
9		vv	. . . . . . . . 9 setvar v
10	9	cv 1394	. . . . . . . 8 class v
11	10	cpw 3649	. . . . . . 7 class ~P v
12	8, 6, 11	crab 2512	. . . . . 6 class { s e. ~P v | E. j j e. s }
13	4, 12, 3	wf1 5318	. . . . 5 wff e : dom e -1-1-> { s e. ~P v | E. j j e. s }
14		vg	. . . . . . 7 setvar g
15	14	cv 1394	. . . . . 6 class g
16		ciedg 15835	. . . . . 6 class iEdg
17	15, 16	cfv 5321	. . . . 5 class (iEdg ` g )
18	13, 2, 17	wsbc 3028	. . . 4 wff [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s }
19		cvtx 15834	. . . . 5 class Vtx
20	15, 19	cfv 5321	. . . 4 class (Vtx ` g )
21	18, 9, 20	wsbc 3028	. . 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 2215	. 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 1395	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  15893