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Theorem mpteq1 4051
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq1  |-  ( A  =  B  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem mpteq1
StepHypRef Expression
1 eqidd 2158 . . 3  |-  ( x  e.  A  ->  C  =  C )
21rgen 2510 . 2  |-  A. x  e.  A  C  =  C
3 mpteq12 4050 . 2  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  C )  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
42, 3mpan2 422 1  |-  ( A  =  B  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128   A.wral 2435    |-> cmpt 4028
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-opab 4029  df-mpt 4030
This theorem is referenced by:  mpteq1d  4052  fmptap  5660  mpompt  5916  mpomptsx  6148  mpompts  6149  tposf12  6219  restco  12670  cnmpt1t  12781  cnmpt2t  12789
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