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Theorem f1oeq1d 41668
Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
f1oeq1d.1 (𝜑𝐹 = 𝐺)
Assertion
Ref Expression
f1oeq1d (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))

Proof of Theorem f1oeq1d
StepHypRef Expression
1 f1oeq1d.1 . 2 (𝜑𝐹 = 𝐺)
2 f1oeq1 6593 . 2 (𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
31, 2syl 17 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐺:𝐴1-1-onto𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  1-1-ontowf1o 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351
This theorem is referenced by: (None)
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