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Theorem fmtnorn 43573
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 7178 . . 3 ((2↑(2↑𝑛)) + 1) ∈ V
2 df-fmtno 43567 . . 3 FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
31, 2fnmpti 6484 . 2 FermatNo Fn ℕ0
4 fvelrnb 6719 . 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 207   = wceq 1528  wcel 2105  wrex 3136  ran crn 5549   Fn wfn 6343  cfv 6348  (class class class)co 7145  1c1 10526   + caddc 10528  2c2 11680  0cn0 11885  cexp 13417  FermatNocfmtno 43566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-ov 7148  df-fmtno 43567
This theorem is referenced by:  prmdvdsfmtnof1lem2  43624  prmdvdsfmtnof  43625  prmdvdsfmtnof1  43626
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