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Theorem foeq2 5337
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 5207 . . 3  |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
21anbi1d 460 . 2  |-  ( A  =  B  ->  (
( F  Fn  A  /\  ran  F  =  C )  <->  ( F  Fn  B  /\  ran  F  =  C ) ) )
3 df-fo 5124 . 2  |-  ( F : A -onto-> C  <->  ( F  Fn  A  /\  ran  F  =  C ) )
4 df-fo 5124 . 2  |-  ( F : B -onto-> C  <->  ( F  Fn  B  /\  ran  F  =  C ) )
52, 3, 43bitr4g 222 1  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   ran crn 4535    Fn wfn 5113   -onto->wfo 5116
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 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-fn 5121  df-fo 5124
This theorem is referenced by:  f1oeq2  5352  foeq123d  5356  tposfo  6161  ctssdclemr  6990  enumct  6993  exmidfodomrlemr  7051  exmidfodomrlemrALT  7052  ctinf  11932  ctiunct  11942  subctctexmid  13185
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