Theorem fmtnorn 47644

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

Step	Hyp	Ref	Expression
1		ovex 7379	. . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V
2		df-fmtno 47638	. . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
3	1, 2	fnmpti 6624	. 2 ⊢ FermatNo Fn ℕ0
4		fvelrnb 6882	. 2 ⊢ (FermatNo Fn ℕ0 → (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹))
5	3, 4	ax-mp 5	1 ⊢ (𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ran crn 5615   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346  1c1 11007   + caddc 11009  2c2 12180  ℕ₀cn0 12381  ↑cexp 13968  FermatNocfmtno 47637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-ov 7349  df-fmtno 47638

This theorem is referenced by:  prmdvdsfmtnof1lem2  47695  prmdvdsfmtnof  47696  prmdvdsfmtnof1  47697