Theorem cbvmptv

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013) Add disjoint variable condition to avoid auxiliary axioms . See cbvmptvg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Nov-2024)

	Ref	Expression
Hypothesis	cbvmptv.1	|- ( x = y -> B = C )
Assertion	cbvmptv	|- ( x e. A |-> B ) = ( y e. A |-> C )

Proof

Step	Hyp	Ref	Expression
1		cbvmptv.1	|- ( x = y -> B = C )
2		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
3	1	eqeq2d	|- ( x = y -> ( z = B <-> z = C ) )
4	2 3	anbi12d	|- ( x = y -> ( ( x e. A /\ z = B ) <-> ( y e. A /\ z = C ) ) )
5	4	cbvopab1v	|- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
6		df-mpt	|- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
7		df-mpt	|- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
8	5 6 7	3eqtr4i	|- ( x e. A |-> B ) = ( y e. A |-> C )