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Theorem f1oeq123d 5066
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 5060 . . 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 5061 . . 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 5062 . . 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 1242  1-1-ontowf1o 4844
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1oprg  5111
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