Theorem uspgrspren

Description: The set G of the "simple pseudographs" with a fixed set of vertices V and the class P of subsets of the set of pairs over the fixed set V are equinumerous. (Contributed by AV, 27-Nov-2021)

	Ref	Expression
Hypotheses	uspgrsprf.p	|- P = ~P ( Pairs ` V )
	uspgrsprf.g	|- G = { <. v , e >. | ( v = V /\ E. q e. USPGraph ( ( Vtx ` q ) = v /\ ( Edg ` q ) = e ) ) }
Assertion	uspgrspren	|- ( V e. W -> G ~~ P )

Proof

Step	Hyp	Ref	Expression
1		uspgrsprf.p	|- P = ~P ( Pairs ` V )
2		uspgrsprf.g	|- G = { <. v , e >. | ( v = V /\ E. q e. USPGraph ( ( Vtx ` q ) = v /\ ( Edg ` q ) = e ) ) }
3	1 2	uspgrbispr	|- ( V e. W -> E. f f : G -1-1-onto-> P )
4		bren	|- ( G ~~ P <-> E. f f : G -1-1-onto-> P )
5	3 4	sylibr	|- ( V e. W -> G ~~ P )