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Theorem foeq3 5182
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 2094 . . 3  |-  ( A  =  B  ->  ( ran  F  =  A  <->  ran  F  =  B ) )
21anbi2d 452 . 2  |-  ( A  =  B  ->  (
( F  Fn  C  /\  ran  F  =  A )  <->  ( F  Fn  C  /\  ran  F  =  B ) ) )
3 df-fo 4978 . 2  |-  ( F : C -onto-> A  <->  ( F  Fn  C  /\  ran  F  =  A ) )
4 df-fo 4978 . 2  |-  ( F : C -onto-> B  <->  ( F  Fn  C  /\  ran  F  =  B ) )
52, 3, 43bitr4g 221 1  |-  ( A  =  B  ->  ( F : C -onto-> A  <->  F : C -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287   ran crn 4405    Fn wfn 4967   -onto->wfo 4970
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 1379  ax-gen 1381  ax-4 1443  ax-17 1462  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-fo 4978
This theorem is referenced by:  f1oeq3  5197  foeq123d  5200  resdif  5226  ffoss  5236  exmidfodomrlemr  6749  exmidfodomrlemrALT  6750
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