Theorem fmtnorn 44986

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 7308	. . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V
2		df-fmtno 44980	. . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
3	1, 2	fnmpti 6576	. 2 ⊢ FermatNo Fn ℕ0
4		fvelrnb 6830	. 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 205   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  ran crn 5590   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  1c1 10872   + caddc 10874  2c2 12028  ℕ₀cn0 12233  ↑cexp 13782  FermatNocfmtno 44979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ov 7278  df-fmtno 44980

This theorem is referenced by:  prmdvdsfmtnof1lem2  45037  prmdvdsfmtnof  45038  prmdvdsfmtnof1  45039