Theorem fmtnorn 42228

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 6910	. . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V
2		df-fmtno 42222	. . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
3	1, 2	fnmpti 6233	. 2 ⊢ FermatNo Fn ℕ0
4		fvelrnb 6468	. 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 198   = wceq 1653   ∈ wcel 2157  ∃wrex 3090  ran crn 5313   Fn wfn 6096  ‘cfv 6101  (class class class)co 6878  1c1 10225   + caddc 10227  2c2 11368  ℕ₀cn0 11580  ↑cexp 13114  FermatNocfmtno 42221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109  df-ov 6881  df-fmtno 42222

This theorem is referenced by:  prmdvdsfmtnof1lem2  42279  prmdvdsfmtnof  42280  prmdvdsfmtnof1  42281