Theorem ello1 15548

Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Assertion

Ref	Expression
ello1	⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚))

Distinct variable group:   𝑥,𝑚,𝑦,𝐹

Proof of Theorem ello1

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5917	. . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
2	1	ineq1d 4227	. . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
3		fveq1 6906	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
4	3	breq1d 5158	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑦) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚))
5	2, 4	raleqbidv 3344	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚))
6	5	2rexbidv 3220	. 2 ⊢ (𝑓 = 𝐹 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚 ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚))
7		df-lo1 15524	. 2 ⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚}
8	6, 7	elrab2 3698	1 ⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ∩ cin 3962   class class class wbr 5148  dom cdm 5689  ‘cfv 6563  (class class class)co 7431   ↑_pm cpm 8866  ℝcr 11152  +∞cpnf 11290   ≤ cle 11294  [,)cico 13386  ≤𝑂(1)clo1 15520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-lo1 15524

This theorem is referenced by:  ello12  15549  lo1f  15551  lo1dm  15552