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Theorem elo1 14464
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 5462 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
21ineq1d 3962 . . . 4 (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
3 fveq1 6331 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
43fveq2d 6336 . . . . 5 (𝑓 = 𝐹 → (abs‘(𝑓𝑦)) = (abs‘(𝐹𝑦)))
54breq1d 4794 . . . 4 (𝑓 = 𝐹 → ((abs‘(𝑓𝑦)) ≤ 𝑚 ↔ (abs‘(𝐹𝑦)) ≤ 𝑚))
62, 5raleqbidv 3300 . . 3 (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹𝑦)) ≤ 𝑚))
762rexbidv 3204 . 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 14428 . 2 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
97, 8elrab2 3516 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 196  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  cin 3720   class class class wbr 4784  dom cdm 5249  cfv 6031  (class class class)co 6792  pm cpm 8009  cc 10135  cr 10136  +∞cpnf 10272  cle 10276  [,)cico 12381  abscabs 14181  𝑂(1)co1 14424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-dm 5259  df-iota 5994  df-fv 6039  df-o1 14428
This theorem is referenced by:  elo12  14465  o1f  14467  o1dm  14468
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