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Theorem foeq123d 6591
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 6568 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
31, 2syl 17 . 2 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 foeq2 6569 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
64, 5syl 17 . 2 (𝜑 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 foeq3 6570 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
97, 8syl 17 . 2 (𝜑 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
103, 6, 93bitrd 308 1 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  ontowfo 6332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-fun 6336  df-fn 6337  df-fo 6340
This theorem is referenced by:  fullfo  17173  cofull  17195  resgrpplusfrn  18108  efabl  25140  iseupth  27984  fundcmpsurinjimaid  43871  fundcmpsurinjALT  43872
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