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Theorem foeq2 5230
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq2  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )

Proof of Theorem foeq2
StepHypRef Expression
1 fneq2 5103 . . 3  |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
21anbi1d 453 . 2  |-  ( A  =  B  ->  (
( F  Fn  A  /\  ran  F  =  C )  <->  ( F  Fn  B  /\  ran  F  =  C ) ) )
3 df-fo 5021 . 2  |-  ( F : A -onto-> C  <->  ( F  Fn  A  /\  ran  F  =  C ) )
4 df-fo 5021 . 2  |-  ( F : B -onto-> C  <->  ( F  Fn  B  /\  ran  F  =  C ) )
52, 3, 43bitr4g 221 1  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   ran crn 4439    Fn wfn 5010   -onto->wfo 5013
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 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-fn 5018  df-fo 5021
This theorem is referenced by:  f1oeq2  5245  foeq123d  5249  tposfo  6036  exmidfodomrlemr  6828  exmidfodomrlemrALT  6829
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