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Mirrors > Home > MPE Home > Th. List > ello1 | Structured version Visualization version GIF version |
Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
ello1 | ⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5772 | . . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
2 | 1 | ineq1d 4188 | . . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞))) |
3 | fveq1 6669 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) | |
4 | 3 | breq1d 5076 | . . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑦) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚)) |
5 | 2, 4 | raleqbidv 3401 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚)) |
6 | 5 | 2rexbidv 3300 | . 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 14848 | . 2 ⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚} | |
8 | 6, 7 | elrab2 3683 | 1 ⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ∩ cin 3935 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ↑pm cpm 8407 ℝcr 10536 +∞cpnf 10672 ≤ cle 10676 [,)cico 12741 ≤𝑂(1)clo1 14844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-dm 5565 df-iota 6314 df-fv 6363 df-lo1 14848 |
This theorem is referenced by: ello12 14873 lo1f 14875 lo1dm 14876 |
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