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Theorem f1oeq123d 5123
 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 5117 . . 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 5118 . . 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 5119 . . 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 203 1 (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  –1-1-onto→wf1o 4901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909 This theorem is referenced by:  f1oprg  5168
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