Theorem opiedgfv

Description: The set of indexed edges 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	opiedgfv	|- ( ( V e. X /\ E e. Y ) -> ( iEdg ` <. V , E >. ) = E )

Proof

Step	Hyp	Ref	Expression
1		opelvvg	|- ( ( V e. X /\ E e. Y ) -> <. V , E >. e. ( _V X. _V ) )
2		opiedgval	|- ( <. V , E >. e. ( _V X. _V ) -> ( iEdg ` <. V , E >. ) = ( 2nd ` <. V , E >. ) )
3	1 2	syl	|- ( ( V e. X /\ E e. Y ) -> ( iEdg ` <. V , E >. ) = ( 2nd ` <. V , E >. ) )
4		op2ndg	|- ( ( V e. X /\ E e. Y ) -> ( 2nd ` <. V , E >. ) = E )
5	3 4	eqtrd	|- ( ( V e. X /\ E e. Y ) -> ( iEdg ` <. V , E >. ) = E )