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Theorem fmtnorn 47458
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 7463 . . 3 ((2↑(2↑𝑛)) + 1) ∈ V
2 df-fmtno 47452 . . 3 FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
31, 2fnmpti 6711 . 2 FermatNo Fn ℕ0
4 fvelrnb 6968 . 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 206   = wceq 1536  wcel 2105  wrex 3067  ran crn 5689   Fn wfn 6557  cfv 6562  (class class class)co 7430  1c1 11153   + caddc 11155  2c2 12318  0cn0 12523  cexp 14098  FermatNocfmtno 47451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-ov 7433  df-fmtno 47452
This theorem is referenced by:  prmdvdsfmtnof1lem2  47509  prmdvdsfmtnof  47510  prmdvdsfmtnof1  47511
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