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Theorem foeq3 5343
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq3  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )

Proof of Theorem foeq3
StepHypRef Expression
1 eqeq2 2149 . . 3  |-  ( A  =  B  ->  ( ran  F  =  A  <->  ran  F  =  B ) )
21anbi2d 459 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  =  A )  <->  ( F  Fn  C  /\  ran  F  =  B ) ) )
3 df-fo 5129 . 2  |-  ( F : C -onto-> A  <->  ( F  Fn  C  /\  ran  F  =  A ) )
4 df-fo 5129 . 2  |-  ( F : C -onto-> B  <->  ( F  Fn  C  /\  ran  F  =  B ) )
52, 3, 43bitr4g 222 1  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   ran crn 4540    Fn wfn 5118   -onto->wfo 5121
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 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-fo 5129
This theorem is referenced by:  f1oeq3  5358  foeq123d  5361  resdif  5389  ffoss  5399  fifo  6868  enumct  7000  ctssexmid  7024  exmidfodomrlemr  7063  exmidfodomrlemrALT  7064  qnnen  11955  ctiunctal  11965  unct  11966  subctctexmid  13255
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