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Theorem foeq1 5553
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 5438 . . 3  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
2 rneq 5007 . . . 4  |-  ( F  =  G  ->  ran  F  =  ran  G )
32eqeq1d 2374 . . 3  |-  ( F  =  G  ->  ( ran  F  =  B  <->  ran  G  =  B ) )
41, 3anbi12d 691 . 2  |-  ( F  =  G  ->  (
( F  Fn  A  /\  ran  F  =  B )  <->  ( G  Fn  A  /\  ran  G  =  B ) ) )
5 df-fo 5364 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
6 df-fo 5364 . 2  |-  ( G : A -onto-> B  <->  ( G  Fn  A  /\  ran  G  =  B ) )
74, 5, 63bitr4g 279 1  |-  ( F  =  G  ->  ( F : A -onto-> B  <->  G : A -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647   ran crn 4793    Fn wfn 5353   -onto->wfo 5356
This theorem is referenced by:  f1oeq1  5569  foeq123d  5574  resdif  5600  exfo  5789  fodomr  7155  fowdom  7432  brwdom2  7434  canthp1lem2  8422  znzrhfo  16718  pjhfo  22598  elunop  22765  elunop2  22906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-fun 5360  df-fn 5361  df-fo 5364
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