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Theorem feq12d 5230
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq12d.1  |-  ( ph  ->  F  =  G )
feq12d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
feq12d  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )

Proof of Theorem feq12d
StepHypRef Expression
1 feq12d.1 . . 3  |-  ( ph  ->  F  =  G )
21feq1d 5227 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : A --> C ) )
3 feq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43feq2d 5228 . 2  |-  ( ph  ->  ( G : A --> C 
<->  G : B --> C ) )
52, 4bitrd 187 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   -->wf 5087
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  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-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-f 5095
This theorem is referenced by:  feq123d  5231  smoeq  6153  lmbr2  12278  lmff  12313  limccl  12671  ellimc3apf  12672
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