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Mirrors > Home > MPE Home > Th. List > ello1d | Structured version Visualization version GIF version |
Description: Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
ello1mpt.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ello1mpt.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
ello1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ello1d.4 | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
ello1d.5 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ≤ 𝑀) |
Ref | Expression |
---|---|
ello1d | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ello1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | ello1d.4 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
3 | ello1d.5 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ≤ 𝑀) | |
4 | 3 | expr 459 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)) |
5 | 4 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)) |
6 | breq1 5061 | . . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥)) | |
7 | 6 | imbi1d 344 | . . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
8 | 7 | ralbidv 3197 | . . . 4 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
9 | breq2 5062 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝐵 ≤ 𝑚 ↔ 𝐵 ≤ 𝑀)) | |
10 | 9 | imbi2d 343 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))) |
11 | 10 | ralbidv 3197 | . . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))) |
12 | 8, 11 | rspc2ev 3634 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)) |
13 | 1, 2, 5, 12 | syl3anc 1367 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)) |
14 | ello1mpt.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
15 | ello1mpt.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
16 | 14, 15 | ello1mpt 14872 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
17 | 13, 16 | mpbird 259 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5058 ↦ cmpt 5138 ℝcr 10530 ≤ cle 10670 ≤𝑂(1)clo1 14838 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ico 12738 df-lo1 14842 |
This theorem is referenced by: elo1d 14887 o1lo12 14889 icco1 14891 lo1const 14971 dirith2 26098 pntrlog2bndlem4 26150 pntrlog2bndlem6 26153 |
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