Theorem fmtnorn 44874

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 7288	. . 3 ⊢ ((2↑(2↑𝑛)) + 1) ∈ V
2		df-fmtno 44868	. . 3 ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))
3	1, 2	fnmpti 6560	. 2 ⊢ FermatNo Fn ℕ0
4		fvelrnb 6812	. 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 2108  ∃wrex 3064  ran crn 5581   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  1c1 10803   + caddc 10805  2c2 11958  ℕ₀cn0 12163  ↑cexp 13710  FermatNocfmtno 44867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-ov 7258  df-fmtno 44868

This theorem is referenced by:  prmdvdsfmtnof1lem2  44925  prmdvdsfmtnof  44926  prmdvdsfmtnof1  44927