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Theorem elbigofrcl 44820
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 6685 . 2 (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ dom Ο)
2 df-bigo 44818 . . . 4 Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})
32dmeqi 5756 . . 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 6079 . . . 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 7173 . . . . . 6 (ℝ ↑pm ℝ) ∈ V
65rabex 5218 . . . . 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 3146 . . 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 2847 . 2 dom Ο = (ℝ ↑pm ℝ)
101, 9eleqtrdi 2926 1 (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3132  wrex 3133  {crab 3136  Vcvv 3479  cin 3917   class class class wbr 5049  cmpt 5129  dom cdm 5538  cfv 6338  (class class class)co 7140  pm cpm 8392  cr 10523   · cmul 10529  +∞cpnf 10659  cle 10663  [,)cico 12728  Οcbigo 44817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-mpt 5130  df-xp 5544  df-rel 5545  df-cnv 5546  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fv 6346  df-ov 7143  df-bigo 44818
This theorem is referenced by:  elbigo  44821  elbigoimp  44826
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