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Theorem f1oeq123d 6090
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 6084 . . 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 6085 . . 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 6086 . . 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 294 1 (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  1-1-ontowf1o 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854
This theorem is referenced by:  f1oprswap  6137  f1oprg  6138  cnfcom  8541  ackbij2lem2  9006  s2f1o  13597  s4f1o  13599  idffth  16514  ressffth  16519  symg1bas  17737  symg2bas  17739  symgfixels  17775  symgfixelsi  17776  rhmf1o  18653  mat1f1o  20203  isismt  25329  ushgredgedg  26014  ushgredgedgloop  26016  trlreslem  26465  wwlksnextbij  26666  eupth0  26940  eupthp1  26942  foresf1o  29190  f1ocnt  29400  indf1ofs  29869  eulerpartgbij  30215  eulerpartlemn  30224  poimirlem16  33057  poimirlem17  33058  poimirlem19  33060  poimirlem20  33061  poimirlem28  33069  wessf1ornlem  38845  disjf1o  38852  ssnnf1octb  38856  sge0fodjrnlem  39940  rnghmf1o  41191
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