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Theorem elo1 15414
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 5860 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
21ineq1d 4172 . . . 4 (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
3 fveq1 6842 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
43fveq2d 6847 . . . . 5 (𝑓 = 𝐹 → (abs‘(𝑓𝑦)) = (abs‘(𝐹𝑦)))
54breq1d 5116 . . . 4 (𝑓 = 𝐹 → ((abs‘(𝑓𝑦)) ≤ 𝑚 ↔ (abs‘(𝐹𝑦)) ≤ 𝑚))
62, 5raleqbidv 3318 . . 3 (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))
762rexbidv 3210 . 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 15378 . 2 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
97, 8elrab2 3649 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 205  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cin 3910   class class class wbr 5106  dom cdm 5634  cfv 6497  (class class class)co 7358  pm cpm 8769  cc 11054  cr 11055  +∞cpnf 11191  cle 11195  [,)cico 13272  abscabs 15125  𝑂(1)co1 15374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-dm 5644  df-iota 6449  df-fv 6505  df-o1 15378
This theorem is referenced by:  elo12  15415  o1f  15417  o1dm  15418
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