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Mirrors > Home > MPE Home > Th. List > ello1mpt | 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 |
---|---|
ello1mpt.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ello1mpt.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
ello1mpt | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ello1mpt.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
2 | 1 | fmpttd 6856 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
3 | ello1mpt.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
4 | ello12 14865 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚))) | |
5 | 2, 3, 4 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚))) |
6 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧 | |
7 | nffvmpt1 6656 | . . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) | |
8 | nfcv 2955 | . . . . . . 7 ⊢ Ⅎ𝑥 ≤ | |
9 | nfcv 2955 | . . . . . . 7 ⊢ Ⅎ𝑥𝑚 | |
10 | 7, 8, 9 | nfbr 5077 | . . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚 |
11 | 6, 10 | nfim 1897 | . . . . 5 ⊢ Ⅎ𝑥(𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) |
12 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) | |
13 | breq2 5034 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥)) | |
14 | fveq2 6645 | . . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) | |
15 | 14 | breq1d 5040 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)) |
16 | 13, 15 | imbi12d 348 | . . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))) |
17 | 11, 12, 16 | cbvralw 3387 | . . . 4 ⊢ (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)) |
18 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
19 | eqid 2798 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
20 | 19 | fvmpt2 6756 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
21 | 18, 1, 20 | syl2anc 587 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
22 | 21 | breq1d 5040 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚 ↔ 𝐵 ≤ 𝑚)) |
23 | 22 | imbi2d 344 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
24 | 23 | ralbidva 3161 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
25 | 17, 24 | syl5bb 286 | . . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
26 | 25 | 2rexbidv 3259 | . 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
27 | 5, 26 | bitrd 282 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 ℝcr 10525 ≤ cle 10665 ≤𝑂(1)clo1 14836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732 df-lo1 14840 |
This theorem is referenced by: ello1mpt2 14871 ello1d 14872 elo1mpt 14883 o1lo1 14886 lo1resb 14913 lo1add 14975 lo1mul 14976 lo1le 15000 |
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