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Theorem feq123 5232
Description: Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
Assertion
Ref Expression
feq123  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  G : C --> D ) )

Proof of Theorem feq123
StepHypRef Expression
1 simp1 964 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  F  =  G )
2 simp2 965 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  A  =  C )
3 simp3 966 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  B  =  D )
41, 2, 3feq123d 5231 1  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  G : C --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 945    = 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: (None)
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