Theorem fmtnorn 41974

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 6823	. . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V
2		df-fmtno 41968	. . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
3	1, 2	fnmpti 6162	. 2 ⊢ FermatNo Fn ℕ0
4		fvelrnb 6385	. 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 196   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  ran crn 5250   Fn wfn 6026  ‘cfv 6031  (class class class)co 6793  1c1 10139   + caddc 10141  2c2 11272  ℕ₀cn0 11494  ↑cexp 13067  FermatNocfmtno 41967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796  df-fmtno 41968

This theorem is referenced by:  prmdvdsfmtnof1lem2  42025  prmdvdsfmtnof  42026  prmdvdsfmtnof1  42027