Theorem cbvmptvw2

Description: Change bound variable and domain in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvmptvw2.1	|- ( x = y -> C = D )
	cbvmptvw2.2	|- ( x = y -> A = B )
Assertion	cbvmptvw2	|- ( x e. A |-> C ) = ( y e. B |-> D )

Proof

Step	Hyp	Ref	Expression
1		cbvmptvw2.1	|- ( x = y -> C = D )
2		cbvmptvw2.2	|- ( x = y -> A = B )
3		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
4	2	eleq2d	|- ( x = y -> ( y e. A <-> y e. B ) )
5	3 4	bitrd	|- ( x = y -> ( x e. A <-> y e. B ) )
6	1	eqeq2d	|- ( x = y -> ( t = C <-> t = D ) )
7	5 6	anbi12d	|- ( x = y -> ( ( x e. A /\ t = C ) <-> ( y e. B /\ t = D ) ) )
8	7	cbvopab1v	|- { <. x , t >. | ( x e. A /\ t = C ) } = { <. y , t >. | ( y e. B /\ t = D ) }
9		df-mpt	|- ( x e. A |-> C ) = { <. x , t >. | ( x e. A /\ t = C ) }
10		df-mpt	|- ( y e. B |-> D ) = { <. y , t >. | ( y e. B /\ t = D ) }
11	8 9 10	3eqtr4i	|- ( x e. A |-> C ) = ( y e. B |-> D )