Theorem bigoval 49132

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[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))})

Distinct variable group:   𝑓,𝐺,𝑥,𝑚,𝑦

Proof of Theorem bigoval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6861	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑦) = (𝐺‘𝑦))
2	1	oveq2d 7407	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑚 · (𝑔‘𝑦)) = (𝑚 · (𝐺‘𝑦)))
3	2	breq2d 5109	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦)) ↔ (𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
4	3	ralbidv 3184	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦)) ↔ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
5	4	2rexbidv 3226	. . 3 ⊢ (𝑔 = 𝐺 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
6	5	rabbidv 3420	. 2 ⊢ (𝑔 = 𝐺 → {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))} = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))})
7		df-bigo 49131	. 2 ⊢ Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))})
8		ovex 7424	. . 3 ⊢ (ℝ ↑pm ℝ) ∈ V
9	8	rabex 5292	. 2 ⊢ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))} ∈ V
10	6, 7, 9	fvmpt 6970	1 ⊢ (𝐺 ∈ (ℝ ↑pm ℝ) → (Ο‘𝐺) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))})

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  {crab 3413   ∩ cin 3901   class class class wbr 5097  dom cdm 5643  ‘cfv 6516  (class class class)co 7391   ↑_pm cpm 8803  ℝcr 11066   · cmul 11072  +∞cpnf 11207   ≤ cle 11211  [,)cico 13345  Οcbigo 49130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-bigo 49131

This theorem is referenced by:  elbigo  49134