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Theorem f1eq123d 6169
Description: Equality deduction for one-to-one 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
f1eq123d (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐵1-1𝐷))

Proof of Theorem f1eq123d
StepHypRef Expression
1 f1eq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 f1eq1 6134 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴1-1𝐶𝐺:𝐴1-1𝐶))
31, 2syl 17 . 2 (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐴1-1𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 f1eq2 6135 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴1-1𝐶𝐺:𝐵1-1𝐶))
64, 5syl 17 . 2 (𝜑 → (𝐺:𝐴1-1𝐶𝐺:𝐵1-1𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 f1eq3 6136 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵1-1𝐶𝐺:𝐵1-1𝐷))
97, 8syl 17 . 2 (𝜑 → (𝐺:𝐵1-1𝐶𝐺:𝐵1-1𝐷))
103, 6, 93bitrd 294 1 (𝜑 → (𝐹:𝐴1-1𝐶𝐺:𝐵1-1𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  1-1wf1 5923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931
This theorem is referenced by:  f10d  6208  fthf1  16624  cofth  16642  istrkgld  25403  istrkg2ld  25404  isushgr  26001  isuspgr  26092  isusgr  26093  isuspgrop  26101  isusgrop  26102  ausgrusgrb  26105  ausgrusgri  26108  usgrstrrepe  26172  uspgr1e  26181  usgrexmpl  26200  usgrres1  26252  usgrexi  26393  uspgr2wlkeq  26598  usgr2trlncl  26712  aciunf1  29591
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