Theorem mpteq12df

Description: An equality inference for the maps-to notation. Compare mpteq12dv . (Contributed by Scott Fenton, 8-Aug-2013) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Hypotheses	mpteq12df.1	|- F/ x ph
	mpteq12df.2	|- ( ph -> A = C )
	mpteq12df.3	|- ( ph -> B = D )
Assertion	mpteq12df	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12df.1	|- F/ x ph
2		mpteq12df.2	|- ( ph -> A = C )
3		mpteq12df.3	|- ( ph -> B = D )
4		nfv	|- F/ y ph
5	2	eleq2d	|- ( ph -> ( x e. A <-> x e. C ) )
6	3	eqeq2d	|- ( ph -> ( y = B <-> y = D ) )
7	5 6	anbi12d	|- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
8	1 4 7	opabbid	|- ( ph -> { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = D ) } )
9		df-mpt	|- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
10		df-mpt	|- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
11	8 9 10	3eqtr4g	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )