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Theorem foeq2 5448
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 5334 . . 3  |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
21anbi1d 685 . 2  |-  ( A  =  B  ->  (
( F  Fn  A  /\  ran  F  =  C )  <->  ( F  Fn  B  /\  ran  F  =  C ) ) )
3 df-fo 5261 . 2  |-  ( F : A -onto-> C  <->  ( F  Fn  A  /\  ran  F  =  C ) )
4 df-fo 5261 . 2  |-  ( F : B -onto-> C  <->  ( F  Fn  B  /\  ran  F  =  C ) )
52, 3, 43bitr4g 279 1  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   ran crn 4690    Fn wfn 5250   -onto->wfo 5253
This theorem is referenced by:  f1oeq2  5464  foeq123d  5468  tposfo  6261  brwdom  7281  brwdom2  7287  canthwdom  7293  cfslb2n  7894  fodomg  8150  0ramcl  13070  ghmcyg  15182  txcmpb  17338  qtoptopon  17395  opidon2  20991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-cleq 2276  df-fn 5258  df-fo 5261
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