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Theorem fmtnorn 43908
 Description: A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.)
Assertion
Ref Expression
fmtnorn (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
Distinct variable group:   𝑛,𝐹

Proof of Theorem fmtnorn
StepHypRef Expression
1 ovex 7173 . . 3 ((2↑(2↑𝑛)) + 1) ∈ V
2 df-fmtno 43902 . . 3 FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
31, 2fnmpti 6474 . 2 FermatNo Fn ℕ0
4 fvelrnb 6709 . 2 (FermatNo Fn ℕ0 → (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹))
53, 4ax-mp 5 1 (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2115  ∃wrex 3133  ran crn 5539   Fn wfn 6333  ‘cfv 6338  (class class class)co 7140  1c1 10525   + caddc 10527  2c2 11680  ℕ0cn0 11885  ↑cexp 13425  FermatNocfmtno 43901 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-iota 6297  df-fun 6340  df-fn 6341  df-fv 6346  df-ov 7143  df-fmtno 43902 This theorem is referenced by:  prmdvdsfmtnof1lem2  43959  prmdvdsfmtnof  43960  prmdvdsfmtnof1  43961
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