ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpteq1 Unicode version

Theorem mpteq1 3980
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 2116 . . 3  |-  ( x  e.  A  ->  C  =  C )
21rgen 2460 . 2  |-  A. x  e.  A  C  =  C
3 mpteq12 3979 . 2  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  C )  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
42, 3mpan2 419 1  |-  ( A  =  B  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   A.wral 2391    |-> cmpt 3957
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2396  df-opab 3958  df-mpt 3959
This theorem is referenced by:  mpteq1d  3981  fmptap  5576  mpompt  5829  mpomptsx  6061  mpompts  6062  tposf12  6132  restco  12238  cnmpt1t  12349  cnmpt2t  12357
  Copyright terms: Public domain W3C validator