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Theorem f1oeq23 6821
Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
Assertion
Ref Expression
f1oeq23 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))

Proof of Theorem f1oeq23
StepHypRef Expression
1 f1oeq2 6819 . 2 (𝐴 = 𝐵 → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐶))
2 f1oeq3 6820 . 2 (𝐶 = 𝐷 → (𝐹:𝐵1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
31, 2sylan9bb 511 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐹:𝐴1-1-onto𝐶𝐹:𝐵1-1-onto𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  1-1-ontowf1o 6539
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-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3954  df-ss 3964  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547
This theorem is referenced by:  f1ofvswap  7299  enfixsn  9077  ackbij2lem2  10231  seqf1o  14005  eulerthlem2  16711  isgim  19130  islmim  20661  fpwrelmapffs  31937  hgt750lemg  33604  poimirlem3  36429  poimirlem15  36441  eldioph2lem1  41431  fundcmpsurbijinj  46013  isomushgr  46429
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