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Theorem elo1 14105
Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
elo1 (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))
Distinct variable group:   𝑥,𝑚,𝑦,𝐹

Proof of Theorem elo1
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dmeq 5246 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
21ineq1d 3775 . . . 4 (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
3 fveq1 6102 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
43fveq2d 6107 . . . . 5 (𝑓 = 𝐹 → (abs‘(𝑓𝑦)) = (abs‘(𝐹𝑦)))
54breq1d 4593 . . . 4 (𝑓 = 𝐹 → ((abs‘(𝑓𝑦)) ≤ 𝑚 ↔ (abs‘(𝐹𝑦)) ≤ 𝑚))
62, 5raleqbidv 3129 . . 3 (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))
762rexbidv 3039 . 2 (𝑓 = 𝐹 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚 ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))
8 df-o1 14069 . 2 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
97, 8elrab2 3333 1 (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  cin 3539   class class class wbr 4583  dom cdm 5038  cfv 5804  (class class class)co 6549  pm cpm 7745  cc 9813  cr 9814  +∞cpnf 9950  cle 9954  [,)cico 12048  abscabs 13822  𝑂(1)co1 14065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-o1 14069
This theorem is referenced by:  elo12  14106  o1f  14108  o1dm  14109
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