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Theorem bigoval 48398
Description: Set of functions of order G(x). (Contributed by AV, 15-May-2020.)
Assertion
Ref Expression
bigoval (𝐺 ∈ (ℝ ↑pm ℝ) → (Ο‘𝐺) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))})
Distinct variable group:   𝑓,𝐺,𝑥,𝑚,𝑦

Proof of Theorem bigoval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6905 . . . . . . 7 (𝑔 = 𝐺 → (𝑔𝑦) = (𝐺𝑦))
21oveq2d 7446 . . . . . 6 (𝑔 = 𝐺 → (𝑚 · (𝑔𝑦)) = (𝑚 · (𝐺𝑦)))
32breq2d 5159 . . . . 5 (𝑔 = 𝐺 → ((𝑓𝑦) ≤ (𝑚 · (𝑔𝑦)) ↔ (𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))))
43ralbidv 3175 . . . 4 (𝑔 = 𝐺 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦)) ↔ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))))
542rexbidv 3219 . . 3 (𝑔 = 𝐺 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))))
65rabbidv 3440 . 2 (𝑔 = 𝐺 → {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))} = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))})
7 df-bigo 48397 . 2 Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})
8 ovex 7463 . . 3 (ℝ ↑pm ℝ) ∈ V
98rabex 5344 . 2 {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))} ∈ V
106, 7, 9fvmpt 7015 1 (𝐺 ∈ (ℝ ↑pm ℝ) → (Ο‘𝐺) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝐺𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  cin 3961   class class class wbr 5147  dom cdm 5688  cfv 6562  (class class class)co 7430  pm cpm 8865  cr 11151   · cmul 11157  +∞cpnf 11289  cle 11293  [,)cico 13385  Οcbigo 48396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-bigo 48397
This theorem is referenced by:  elbigo  48400
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