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Theorem mpteq2ia 4022
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpteq2ia.1  |-  ( x  e.  A  ->  B  =  C )
Assertion
Ref Expression
mpteq2ia  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C )

Proof of Theorem mpteq2ia
StepHypRef Expression
1 eqid 2140 . . 3  |-  A  =  A
21ax-gen 1426 . 2  |-  A. x  A  =  A
3 mpteq2ia.1 . . 3  |-  ( x  e.  A  ->  B  =  C )
43rgen 2488 . 2  |-  A. x  e.  A  B  =  C
5 mpteq12f 4016 . 2  |-  ( ( A. x  A  =  A  /\  A. x  e.  A  B  =  C )  ->  (
x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
62, 4, 5mp2an 423 1  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330    = wceq 1332    e. wcel 1481   A.wral 2417    |-> cmpt 3997
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-opab 3998  df-mpt 3999
This theorem is referenced by:  mpteq2i  4023  feqresmpt  5483  elfvmptrab  5524  fmptap  5618  offres  6041  cnrecnv  10714  ege2le3  11414  eirraplem  11519  cnmpt1st  12496  cnmpt2nd  12497  expcncf  12800  dvexp  12883  dveflem  12895  dvef  12896
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