Theorem uspgrloopvtx

Description: The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop ). (Contributed by AV, 17-Dec-2020)

	Ref	Expression
Hypothesis	uspgrloopvtx.g	|- G = <. V , { <. A , { N } >. } >.
Assertion	uspgrloopvtx	|- ( V e. W -> ( Vtx ` G ) = V )

Proof

Step	Hyp	Ref	Expression
1		uspgrloopvtx.g	|- G = <. V , { <. A , { N } >. } >.
2	1	fveq2i	|- ( Vtx ` G ) = ( Vtx ` <. V , { <. A , { N } >. } >. )
3		snex	|- { <. A , { N } >. } e. _V
4		opvtxfv	|- ( ( V e. W /\ { <. A , { N } >. } e. _V ) -> ( Vtx ` <. V , { <. A , { N } >. } >. ) = V )
5	3 4	mpan2	|- ( V e. W -> ( Vtx ` <. V , { <. A , { N } >. } >. ) = V )
6	2 5	syl5eq	|- ( V e. W -> ( Vtx ` G ) = V )