Theorem struct2griedg

Description: The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020) (Proof shortened by AV, 12-Nov-2021)

	Ref	Expression
Hypothesis	struct2grvtx.g	|- G = { <. ( Base ` ndx ) , V >. , <. ( .ef ` ndx ) , E >. }
Assertion	struct2griedg	|- ( ( V e. X /\ E e. Y ) -> ( iEdg ` G ) = E )

Proof

Step	Hyp	Ref	Expression
1		struct2grvtx.g	|- G = { <. ( Base ` ndx ) , V >. , <. ( .ef ` ndx ) , E >. }
2	1	struct2grstr	|- G Struct <. ( Base ` ndx ) , ( .ef ` ndx ) >.
3	2	a1i	|- ( ( V e. X /\ E e. Y ) -> G Struct <. ( Base ` ndx ) , ( .ef ` ndx ) >. )
4		simpl	|- ( ( V e. X /\ E e. Y ) -> V e. X )
5		simpr	|- ( ( V e. X /\ E e. Y ) -> E e. Y )
6	1	eqimss2i	|- { <. ( Base ` ndx ) , V >. , <. ( .ef ` ndx ) , E >. } C_ G
7	6	a1i	|- ( ( V e. X /\ E e. Y ) -> { <. ( Base ` ndx ) , V >. , <. ( .ef ` ndx ) , E >. } C_ G )
8	3 4 5 7	structgrssiedg	|- ( ( V e. X /\ E e. Y ) -> ( iEdg ` G ) = E )