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Theorem f1oeq1d 6829
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 6822 . 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 205   = wceq 1542  1-1-ontowf1o 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by:  f1orescnv  6849  f1osng  6875  f1ofvswap  7304  dif1en  9160  dif1enOLD  9162  cnfcomlem  9694  cnfcom2  9697  cnfcom3clem  9700  infxpenc  10013  infxpenc2lem2  10015  infxpenc2  10017  canthp1lem2  10648  pwfseqlem5  10658  pwfseq  10659  s2f1o  14867  s4f1o  14869  bitsf1ocnv  16385  yonffthlem  18235  grplactcnv  18926  eqgen  19061  znunithash  21120  tgpconncompeqg  23616  fcobijfs  31948  s2f1  32111  mgcf1o  32173  gsummpt2d  32201  indf1o  33022  subfacp1lem3  34173  subfacp1lem5  34175  ismrer1  36706  hvmap1o  40634  metakunt34  41018
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