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Theorem f1oeq123d 5154
Description: Equality deduction for one-to-one 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
f1oeq123d (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))

Proof of Theorem f1oeq123d
StepHypRef Expression
1 f1eq123d.1 . . 3 (𝜑𝐹 = 𝐺)
2 f1oeq1 5148 . . 3 (𝐹 = 𝐺 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐴1-1-onto𝐶))
31, 2syl 14 . 2 (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐴1-1-onto𝐶))
4 f1eq123d.2 . . 3 (𝜑𝐴 = 𝐵)
5 f1oeq2 5149 . . 3 (𝐴 = 𝐵 → (𝐺:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐶))
64, 5syl 14 . 2 (𝜑 → (𝐺:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐶))
7 f1eq123d.3 . . 3 (𝜑𝐶 = 𝐷)
8 f1oeq3 5150 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
97, 8syl 14 . 2 (𝜑 → (𝐺:𝐵1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
103, 6, 93bitrd 212 1 (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  1-1-ontowf1o 4931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939
This theorem is referenced by:  f1oprg  5199
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