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Theorem foeq1 5678
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq1  |-  ( F  =  G  ->  ( F : A -onto-> B  <->  G : A -onto-> B ) )

Proof of Theorem foeq1
StepHypRef Expression
1 fneq1 5563 . . 3  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
2 rneq 5124 . . . 4  |-  ( F  =  G  ->  ran  F  =  ran  G )
32eqeq1d 2450 . . 3  |-  ( F  =  G  ->  ( ran  F  =  B  <->  ran  G  =  B ) )
41, 3anbi12d 693 . 2  |-  ( F  =  G  ->  (
( F  Fn  A  /\  ran  F  =  B )  <->  ( G  Fn  A  /\  ran  G  =  B ) ) )
5 df-fo 5489 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
6 df-fo 5489 . 2  |-  ( G : A -onto-> B  <->  ( G  Fn  A  /\  ran  G  =  B ) )
74, 5, 63bitr4g 281 1  |-  ( F  =  G  ->  ( F : A -onto-> B  <->  G : A -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653   ran crn 4908    Fn wfn 5478   -onto->wfo 5481
This theorem is referenced by:  f1oeq1  5694  foeq123d  5699  resdif  5725  exfo  5916  fodomr  7287  fowdom  7568  brwdom2  7570  canthp1lem2  8559  znzrhfo  16859  pjhfo  23239  elunop  23406  elunop2  23547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-fun 5485  df-fn 5486  df-fo 5489
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