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Theorem f1oeq123d 6779
Description: Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
f1eq123d.1 (𝜑𝐹 = 𝐺)
f1eq123d.2 (𝜑𝐴 = 𝐵)
f1eq123d.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
f1oeq123d (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))

Proof of Theorem f1oeq123d
StepHypRef Expression
1 f1eq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 f1oeq1 6773 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐴1-1-onto𝐶))
31, 2syl 17 . 2 (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐴1-1-onto𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 f1oeq2 6774 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐶))
64, 5syl 17 . 2 (𝜑 → (𝐺:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 f1oeq3 6775 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
97, 8syl 17 . 2 (𝜑 → (𝐺:𝐵1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
103, 6, 93bitrd 305 1 (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  1-1-ontowf1o 6496
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  f1oprswap  6829  f1oprg  6830  f1ossf1o  7075  cnfcom  9637  ackbij2lem2  10177  idffth  17821  ressffth  17826  symgval  19151  symg1bas  19173  symg2bas  19175  symgfixels  19217  symgfixelsi  19218  rhmf1o  20165  mat1f1o  21830  ushgredgedg  28180  ushgredgedgloop  28182  trlreslem  28650  wlknwwlksnbij  28836  wwlksnextbij  28850  clwlknf1oclwwlkn  29031  eupth0  29161  eupthp1  29163  foresf1o  31434  f1ocnt  31708  symgcom  31937  cycpmcl  31968  cycpmconjslem2  32007  nsgqusf1o  32197  dimkerim  32325  indf1ofs  32628  eulerpartgbij  32975  eulerpartlemn  32984  reprpmtf1o  33242  poimirlem16  36097  poimirlem17  36098  poimirlem19  36100  poimirlem20  36101  poimirlem28  36109  metakunt17  40596  wessf1ornlem  43410  disjf1o  43417  ssnnf1octb  43421  sge0fodjrnlem  44664  f1oresf1orab  45528  isomgr  46022  rnghmf1o  46208
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