Theorem opvtxfv

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020)

	Ref	Expression
Assertion	opvtxfv	|- ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = V )

Proof

Step	Hyp	Ref	Expression
1		opelvvg	|- ( ( V e. X /\ E e. Y ) -> <. V , E >. e. ( _V X. _V ) )
2		opvtxval	|- ( <. V , E >. e. ( _V X. _V ) -> ( Vtx ` <. V , E >. ) = ( 1st ` <. V , E >. ) )
3	1 2	syl	|- ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = ( 1st ` <. V , E >. ) )
4		op1stg	|- ( ( V e. X /\ E e. Y ) -> ( 1st ` <. V , E >. ) = V )
5	3 4	eqtrd	|- ( ( V e. X /\ E e. Y ) -> ( Vtx ` <. V , E >. ) = V )