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Theorem mpteq1 4073
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 2171 . . 3  |-  ( x  e.  A  ->  C  =  C )
21rgen 2523 . 2  |-  A. x  e.  A  C  =  C
3 mpteq12 4072 . 2  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  C )  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
42, 3mpan2 423 1  |-  ( A  =  B  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   A.wral 2448    |-> cmpt 4050
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-opab 4051  df-mpt 4052
This theorem is referenced by:  mpteq1d  4074  fmptap  5686  mpompt  5945  mpomptsx  6176  mpompts  6177  tposf12  6248  restco  12968  cnmpt1t  13079  cnmpt2t  13087
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