Theorem mpteq1d 4067

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)

Hypothesis

Ref	Expression
mpteq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
mpteq1d	|- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem mpteq1d

Step	Hyp	Ref	Expression
1		mpteq1d.1	. 2 |- ( ph -> A = B )
2		mpteq1 4066	. 2 |- ( A = B -> ( x e. A |-> C ) = ( x e. B |-> C ) )
3	1, 2	syl 14	1 |- ( ph -> ( x e. A |-> C ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1343   |-> cmpt 4043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-opab 4044  df-mpt 4045

This theorem is referenced by:  fmptapd  5676  offval  6057