Theorem bigoval 49329

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 6880	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑦) = (𝐺‘𝑦))
2	1	oveq2d 7426	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑚 · (𝑔‘𝑦)) = (𝑚 · (𝐺‘𝑦)))
3	2	breq2d 5121	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦)) ↔ (𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
4	3	ralbidv 3188	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦)) ↔ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))
5	4	2rexbidv 3230	. . 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 3423	. 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 49328	. 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 7443	. . 3 ⊢ (ℝ ↑pm ℝ) ∈ V
9	8	rabex 5309	. 2 ⊢ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))} ∈ V
10	6, 7, 9	fvmpt 6989	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 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416   ∩ cin 3904   class class class wbr 5109  dom cdm 5661  ‘cfv 6536  (class class class)co 7410   ↑_pm cpm 8821  ℝcr 11094   · cmul 11100  +∞cpnf 11235   ≤ cle 11239  [,)cico 13369  Οcbigo 49327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-bigo 49328

This theorem is referenced by:  elbigo  49331