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Theorem feq123 5313
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 982 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  F  =  G )
2 simp2 983 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  A  =  C )
3 simp3 984 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  B  =  D )
41, 2, 3feq123d 5312 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 963    = wceq 1335   -->wf 5168
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-fun 5174  df-fn 5175  df-f 5176
This theorem is referenced by: (None)
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