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Theorem foeq123d 6832
Description: Equality deduction for 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
foeq123d (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))

Proof of Theorem foeq123d
StepHypRef Expression
1 f1eq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 foeq1 6807 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
31, 2syl 17 . 2 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 foeq2 6808 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
64, 5syl 17 . 2 (𝜑 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 foeq3 6809 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
97, 8syl 17 . 2 (𝜑 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
103, 6, 93bitrd 305 1 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  ontowfo 6546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-fun 6550  df-fn 6551  df-fo 6554
This theorem is referenced by:  fullfo  17900  cofull  17922  resgrpplusfrn  18906  efabl  26483  iseupth  30010  funfocofob  46458  fundcmpsurinjimaid  46751  fundcmpsurinjALT  46752
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