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Theorem ello1 14094
Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
ello1 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
Distinct variable group:   𝑥,𝑚,𝑦,𝐹

Proof of Theorem ello1
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 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
43breq1d 4593 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑦) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
52, 4raleqbidv 3129 . . 3 (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
652rexbidv 3039 . 2 (𝑓 = 𝐹 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚 ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
7 df-lo1 14070 . 2 ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚}
86, 7elrab2 3333 1 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(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  cr 9814  +∞cpnf 9950  cle 9954  [,)cico 12048  ≤𝑂(1)clo1 14066
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-lo1 14070
This theorem is referenced by:  ello12  14095  lo1f  14097  lo1dm  14098
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