Theorem opfusgr

Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020)

	Ref	Expression
Assertion	opfusgr	|- ( ( V e. X /\ E e. Y ) -> ( <. V , E >. e. FinUSGraph <-> ( <. V , E >. e. USGraph /\ V e. Fin ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( Vtx ` <. V , E >. ) = ( Vtx ` <. V , E >. )
2	1	isfusgr	|- ( <. V , E >. e. FinUSGraph <-> ( <. V , E >. e. USGraph /\ ( Vtx ` <. V , E >. ) e. Fin ) )
3		opvtxfv	|- ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = V )
4	3	eleq1d	|- ( ( V e. X /\ E e. Y ) -> ( ( Vtx ` <. V , E >. ) e. Fin <-> V e. Fin ) )
5	4	anbi2d	|- ( ( V e. X /\ E e. Y ) -> ( ( <. V , E >. e. USGraph /\ ( Vtx ` <. V , E >. ) e. Fin ) <-> ( <. V , E >. e. USGraph /\ V e. Fin ) ) )
6	2 5	syl5bb	|- ( ( V e. X /\ E e. Y ) -> ( <. V , E >. e. FinUSGraph <-> ( <. V , E >. e. USGraph /\ V e. Fin ) ) )