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Theorem ello1 14731
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[,)+∞))(𝐹𝑦) ≤ 𝑚))
Distinct variable group:   𝑥,𝑚,𝑦,𝐹

Proof of Theorem ello1
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 dmeq 5618 . . . . 5 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
21ineq1d 4069 . . . 4 (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
3 fveq1 6495 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
43breq1d 4935 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑦) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
52, 4raleqbidv 3334 . . 3 (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
652rexbidv 3238 . 2 (𝑓 = 𝐹 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚 ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
7 df-lo1 14707 . 2 ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ 𝑚}
86, 7elrab2 3592 1 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1508  wcel 2051  wral 3081  wrex 3082  cin 3821   class class class wbr 4925  dom cdm 5403  cfv 6185  (class class class)co 6974  pm cpm 8205  cr 10332  +∞cpnf 10469  cle 10473  [,)cico 12554  ≤𝑂(1)clo1 14703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-dm 5413  df-iota 6149  df-fv 6193  df-lo1 14707
This theorem is referenced by:  ello12  14732  lo1f  14734  lo1dm  14735
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