Definition df-ushgr

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)

	Ref	Expression
Assertion	df-ushgr	|- USHGraph = { g | [. ( Vtx ` g ) / v ]. [. ( iEdg ` g ) / e ]. e : dom e -1-1-> ( ~P v \ { (/) } ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cushgr	|- USHGraph
1		vg	|- g
2		cvtx	|- Vtx
3	1	cv	|- g
4	3 2	cfv	|- ( Vtx ` g )
5		vv	|- v
6		ciedg	|- iEdg
7	3 6	cfv	|- ( iEdg ` g )
8		ve	|- e
9	8	cv	|- e
10	9	cdm	|- dom e
11	5	cv	|- v
12	11	cpw	|- ~P v
13		c0	|- (/)
14	13	csn	|- { (/) }
15	12 14	cdif	|- ( ~P v \ { (/) } )
16	10 15 9	wf1	|- e : dom e -1-1-> ( ~P v \ { (/) } )
17	16 8 7	wsbc	|- [. ( iEdg ` g ) / e ]. e : dom e -1-1-> ( ~P v \ { (/) } )
18	17 5 4	wsbc	|- [. ( Vtx ` g ) / v ]. [. ( iEdg ` g ) / e ]. e : dom e -1-1-> ( ~P v \ { (/) } )
19	18 1	cab	|- { g | [. ( Vtx ` g ) / v ]. [. ( iEdg ` g ) / e ]. e : dom e -1-1-> ( ~P v \ { (/) } ) }
20	0 19	wceq	|- USHGraph = { g | [. ( Vtx ` g ) / v ]. [. ( iEdg ` g ) / e ]. e : dom e -1-1-> ( ~P v \ { (/) } ) }