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Theorem foeq1 5229
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 5102 . . 3  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
2 rneq 4662 . . . 4  |-  ( F  =  G  ->  ran  F  =  ran  G )
32eqeq1d 2096 . . 3  |-  ( F  =  G  ->  ( ran  F  =  B  <->  ran  G  =  B ) )
41, 3anbi12d 457 . 2  |-  ( F  =  G  ->  (
( F  Fn  A  /\  ran  F  =  B )  <->  ( G  Fn  A  /\  ran  G  =  B ) ) )
5 df-fo 5021 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
6 df-fo 5021 . 2  |-  ( G : A -onto-> B  <->  ( G  Fn  A  /\  ran  G  =  B ) )
74, 5, 63bitr4g 221 1  |-  ( F  =  G  ->  ( F : A -onto-> B  <->  G : A -onto-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   ran crn 4439    Fn wfn 5010   -onto->wfo 5013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-fo 5021
This theorem is referenced by:  f1oeq1  5244  foeq123d  5249  resdif  5275  dif1en  6593  exmidfodomrlemim  6825
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