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Theorem elbigofrcl 48471
Description: Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.)
Assertion
Ref Expression
elbigofrcl (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))

Proof of Theorem elbigofrcl
Dummy variables 𝑔 𝑓 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6943 . 2 (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ dom Ο)
2 df-bigo 48469 . . . 4 Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})
32dmeqi 5915 . . 3 dom Ο = dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})
4 dmmptg 6262 . . . 4 (∀𝑔 ∈ (ℝ ↑pm ℝ){𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))} ∈ V → dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))}) = (ℝ ↑pm ℝ))
5 ovex 7464 . . . . . 6 (ℝ ↑pm ℝ) ∈ V
65rabex 5339 . . . . 5 {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))} ∈ V
76a1i 11 . . . 4 (𝑔 ∈ (ℝ ↑pm ℝ) → {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))} ∈ V)
84, 7mprg 3067 . . 3 dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))}) = (ℝ ↑pm ℝ)
93, 8eqtri 2765 . 2 dom Ο = (ℝ ↑pm ℝ)
101, 9eleqtrdi 2851 1 (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950   class class class wbr 5143  cmpt 5225  dom cdm 5685  cfv 6561  (class class class)co 7431  pm cpm 8867  cr 11154   · cmul 11160  +∞cpnf 11292  cle 11296  [,)cico 13389  Οcbigo 48468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-bigo 48469
This theorem is referenced by:  elbigo  48472  elbigoimp  48477
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