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Theorem foeq2 5775
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 5651 . . 3  |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
21anbi1d 703 . 2  |-  ( A  =  B  ->  (
( F  Fn  A  /\  ran  F  =  C )  <->  ( F  Fn  B  /\  ran  F  =  C ) ) )
3 df-fo 5575 . 2  |-  ( F : A -onto-> C  <->  ( F  Fn  A  /\  ran  F  =  C ) )
4 df-fo 5575 . 2  |-  ( F : B -onto-> C  <->  ( F  Fn  B  /\  ran  F  =  C ) )
52, 3, 43bitr4g 288 1  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   ran crn 4824    Fn wfn 5564   -onto->wfo 5567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-cleq 2394  df-fn 5572  df-fo 5575
This theorem is referenced by:  f1oeq2  5791  foeq123d  5795  tposfo  6985  brwdom  8027  brwdom2  8033  canthwdom  8039  cfslb2n  8680  fodomg  8935  0ramcl  14750  ghmcyg  17222  txcmpb  20437  qtoptopon  20497  opidon2OLD  25740
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