Theorem mpoeq3ia

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

	Ref	Expression
Hypothesis	mpoeq3ia.1	|- ( ( x e. A /\ y e. B ) -> C = D )
Assertion	mpoeq3ia	|- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )

Proof

Step	Hyp	Ref	Expression
1		mpoeq3ia.1	|- ( ( x e. A /\ y e. B ) -> C = D )
2	1	3adant1	|- ( ( T. /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva	|- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )
4	3	mptru	|- ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D )