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Theorem mpteq1 3882
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 2084 . . 3  |-  ( x  e.  A  ->  C  =  C )
21rgen 2421 . 2  |-  A. x  e.  A  C  =  C
3 mpteq12 3881 . 2  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  C )  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
42, 3mpan2 416 1  |-  ( A  =  B  ->  (
x  e.  A  |->  C )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   A.wral 2353    |-> cmpt 3859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-opab 3860  df-mpt 3861
This theorem is referenced by:  mpteq1d  3883  fmptap  5405  mpt2mpt  5647  mpt2mptsx  5874  mpt2mpts  5875  tposf12  5938
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