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Theorem foeq1 5616
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 5501 . . 3  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
2 rneq 5062 . . . 4  |-  ( F  =  G  ->  ran  F  =  ran  G )
32eqeq1d 2420 . . 3  |-  ( F  =  G  ->  ( ran  F  =  B  <->  ran  G  =  B ) )
41, 3anbi12d 692 . 2  |-  ( F  =  G  ->  (
( F  Fn  A  /\  ran  F  =  B )  <->  ( G  Fn  A  /\  ran  G  =  B ) ) )
5 df-fo 5427 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
6 df-fo 5427 . 2  |-  ( G : A -onto-> B  <->  ( G  Fn  A  /\  ran  G  =  B ) )
74, 5, 63bitr4g 280 1  |-  ( F  =  G  ->  ( F : A -onto-> B  <->  G : A -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   ran crn 4846    Fn wfn 5416   -onto->wfo 5419
This theorem is referenced by:  f1oeq1  5632  foeq123d  5637  resdif  5663  exfo  5854  fodomr  7225  fowdom  7503  brwdom2  7505  canthp1lem2  8492  znzrhfo  16791  pjhfo  23169  elunop  23336  elunop2  23477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-fun 5423  df-fn 5424  df-fo 5427
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