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Theorem foeq1 5803
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 5679 . . 3  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
2 rneq 5076 . . . 4  |-  ( F  =  G  ->  ran  F  =  ran  G )
32eqeq1d 2424 . . 3  |-  ( F  =  G  ->  ( ran  F  =  B  <->  ran  G  =  B ) )
41, 3anbi12d 715 . 2  |-  ( F  =  G  ->  (
( F  Fn  A  /\  ran  F  =  B )  <->  ( G  Fn  A  /\  ran  G  =  B ) ) )
5 df-fo 5604 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
6 df-fo 5604 . 2  |-  ( G : A -onto-> B  <->  ( G  Fn  A  /\  ran  G  =  B ) )
74, 5, 63bitr4g 291 1  |-  ( F  =  G  ->  ( F : A -onto-> B  <->  G : A -onto-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   ran crn 4851    Fn wfn 5593   -onto->wfo 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-fun 5600  df-fn 5601  df-fo 5604
This theorem is referenced by:  f1oeq1  5819  foeq123d  5824  resdif  5848  exfo  6052  fodomr  7726  fowdom  8089  brwdom2  8091  canthp1lem2  9079  mndfo  16549  znzrhfo  19105  pjhfo  27345  elunop  27511  elunop2  27652  nnfoctbdjlem  38072
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