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Theorem f1oeq123d 5332
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 5326 . . 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 5327 . . 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 5328 . . 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 213 1 (𝜑 → (𝐹:𝐴1-1-onto𝐶𝐺:𝐵1-1-onto𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  1-1-ontowf1o 5092
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-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  f1oprg  5379  ennnfonelemhf1o  11853
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