Theorem mpoeq12

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013)

	Ref	Expression
Assertion	mpoeq12	|- ( ( A = C /\ B = D ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- E = E
2	1	rgenw	|- A. y e. B E = E
3	2	jctr	|- ( B = D -> ( B = D /\ A. y e. B E = E ) )
4	3	ralrimivw	|- ( B = D -> A. x e. A ( B = D /\ A. y e. B E = E ) )
5		mpoeq123	|- ( ( A = C /\ A. x e. A ( B = D /\ A. y e. B E = E ) ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )
6	4 5	sylan2	|- ( ( A = C /\ B = D ) -> ( x e. A , y e. B |-> E ) = ( x e. C , y e. D |-> E ) )