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Theorem foeq123d 5331
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 5311 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
31, 2syl 14 . 2 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐴onto𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 foeq2 5312 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
64, 5syl 14 . 2 (𝜑 → (𝐺:𝐴onto𝐶𝐺:𝐵onto𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 foeq3 5313 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
97, 8syl 14 . 2 (𝜑 → (𝐺:𝐵onto𝐶𝐺:𝐵onto𝐷))
103, 6, 93bitrd 213 1 (𝜑 → (𝐹:𝐴onto𝐶𝐺:𝐵onto𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  ontowfo 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-fo 5099
This theorem is referenced by:  ctssexmid  6992  unct  11881
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