Theorem uspgrsprf1o

Description: The mapping F is a bijection between the "simple pseudographs" with a fixed set of vertices V and the subsets of the set of pairs over the set V . See also the comments on uspgrbisymrel . (Contributed by AV, 25-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 ) ) }
	uspgrsprf.f	|- F = ( g e. G |-> ( 2nd ` g ) )
Assertion	uspgrsprf1o	|- ( V e. W -> F : G -1-1-onto-> 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		uspgrsprf.f	|- F = ( g e. G |-> ( 2nd ` g ) )
4	1 2 3	uspgrsprf1	|- F : G -1-1-> P
5	4	a1i	|- ( V e. W -> F : G -1-1-> P )
6	1 2 3	uspgrsprfo	|- ( V e. W -> F : G -onto-> P )
7		df-f1o	|- ( F : G -1-1-onto-> P <-> ( F : G -1-1-> P /\ F : G -onto-> P ) )
8	5 6 7	sylanbrc	|- ( V e. W -> F : G -1-1-onto-> P )