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Theorem foeq123d 6245
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 6224 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
31, 2syl 17 . 2 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 foeq2 6225 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
64, 5syl 17 . 2 (𝜑 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 foeq3 6226 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
97, 8syl 17 . 2 (𝜑 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
103, 6, 93bitrd 294 1 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1596  ontowfo 5999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-fun 6003  df-fn 6004  df-fo 6007
This theorem is referenced by:  fullfo  16694  cofull  16716  resgrpplusfrn  17558  efabl  24416  iseupth  27274
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