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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 456 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)) |
| 5 | 4 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)) |
| 6 | breq1 5146 | . . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥)) | |
| 7 | 6 | imbi1d 341 | . . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
| 8 | 7 | ralbidv 3178 | . . . 4 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
| 9 | breq2 5147 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝐵 ≤ 𝑚 ↔ 𝐵 ≤ 𝑀)) | |
| 10 | 9 | imbi2d 340 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))) |
| 11 | 10 | ralbidv 3178 | . . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))) |
| 12 | 8, 11 | rspc2ev 3635 | . . 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 1373 | . 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 15557 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
| 17 | 13, 16 | mpbird 257 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ℝcr 11154 ≤ cle 11296 ≤𝑂(1)clo1 15523 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 df-lo1 15527 |
| This theorem is referenced by: elo1d 15572 o1lo12 15574 icco1 15576 lo1const 15657 dirith2 27572 pntrlog2bndlem4 27624 pntrlog2bndlem6 27627 |
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