Theorem mpteq12da

Description: An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021) Remove dependency on ax-10 . (Revised by SN, 11-Nov-2024)

	Ref	Expression
Hypotheses	mpteq12da.1	|- F/ x ph
	mpteq12da.2	|- ( ph -> A = C )
	mpteq12da.3	|- ( ( ph /\ x e. A ) -> B = D )
Assertion	mpteq12da	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )

Proof

Step	Hyp	Ref	Expression
1		mpteq12da.1	|- F/ x ph
2		mpteq12da.2	|- ( ph -> A = C )
3		mpteq12da.3	|- ( ( ph /\ x e. A ) -> B = D )
4		nfv	|- F/ y ph
5	3	eqeq2d	|- ( ( ph /\ x e. A ) -> ( y = B <-> y = D ) )
6	5	pm5.32da	|- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. A /\ y = D ) ) )
7	2	eleq2d	|- ( ph -> ( x e. A <-> x e. C ) )
8	7	anbi1d	|- ( ph -> ( ( x e. A /\ y = D ) <-> ( x e. C /\ y = D ) ) )
9	6 8	bitrd	|- ( ph -> ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = D ) ) )
10	1 4 9	opabbid	|- ( ph -> { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = D ) } )
11		df-mpt	|- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
12		df-mpt	|- ( x e. C |-> D ) = { <. x , y >. | ( x e. C /\ y = D ) }
13	10 11 12	3eqtr4g	|- ( ph -> ( x e. A |-> B ) = ( x e. C |-> D ) )