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Metamath Proof Explorer

Theorem cbvmpov

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt , some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013)

Ref Expression
Hypotheses cbvmpov.1 
|- ( x = z -> C = E )
cbvmpov.2 
|- ( y = w -> E = D )
Assertion cbvmpov 
|- ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )

Proof

Step Hyp Ref Expression
1 cbvmpov.1 
 |-  ( x = z -> C = E )
2 cbvmpov.2 
 |-  ( y = w -> E = D )
3 eleq1w 
 |-  ( x = z -> ( x e. A <-> z e. A ) )
4 eleq1w 
 |-  ( y = w -> ( y e. B <-> w e. B ) )
5 3 4 bi2anan9 
 |-  ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. B ) ) )
6 1 2 sylan9eq 
 |-  ( ( x = z /\ y = w ) -> C = D )
7 6 eqeq2d 
 |-  ( ( x = z /\ y = w ) -> ( v = C <-> v = D ) )
8 5 7 anbi12d 
 |-  ( ( x = z /\ y = w ) -> ( ( ( x e. A /\ y e. B ) /\ v = C ) <-> ( ( z e. A /\ w e. B ) /\ v = D ) ) )
9 8 cbvoprab12v 
 |-  { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) } = { <. <. z , w >. , v >. | ( ( z e. A /\ w e. B ) /\ v = D ) }
10 df-mpo 
 |-  ( x e. A , y e. B |-> C ) = { <. <. x , y >. , v >. | ( ( x e. A /\ y e. B ) /\ v = C ) }
11 df-mpo 
 |-  ( z e. A , w e. B |-> D ) = { <. <. z , w >. , v >. | ( ( z e. A /\ w e. B ) /\ v = D ) }
12 9 10 11 3eqtr4i 
 |-  ( x e. A , y e. B |-> C ) = ( z e. A , w e. B |-> D )