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Theorem f1oeq1d 37449
Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
f1oeq1d.1  |-  ( ph  ->  F  =  G )
Assertion
Ref Expression
f1oeq1d  |-  ( ph  ->  ( F : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )

Proof of Theorem f1oeq1d
StepHypRef Expression
1 f1oeq1d.1 . 2  |-  ( ph  ->  F  =  G )
2 f1oeq1 5805 . 2  |-  ( F  =  G  ->  ( F : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
31, 2syl 17 1  |-  ( ph  ->  ( F : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    = wceq 1444   -1-1-onto->wf1o 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589
This theorem is referenced by: (None)
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