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| Mirrors > Home > MPE Home > Th. List > axdc3lem | Structured version Visualization version GIF version | ||
| Description: The class 𝑆 of finite approximations to the DC sequence is a set. (We derive here the stronger statement that 𝑆 is a subset of a specific set, namely 𝒫 (ω × 𝐴).) (Contributed by Mario Carneiro, 27-Jan-2013.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 18-Mar-2014.) | 
| Ref | Expression | 
|---|---|
| axdc3lem.1 | ⊢ 𝐴 ∈ V | 
| axdc3lem.2 | ⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))} | 
| Ref | Expression | 
|---|---|
| axdc3lem | ⊢ 𝑆 ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dcomex 10488 | . . . 4 ⊢ ω ∈ V | |
| 2 | axdc3lem.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 3 | 1, 2 | xpex 7774 | . . 3 ⊢ (ω × 𝐴) ∈ V | 
| 4 | 3 | pwex 5379 | . 2 ⊢ 𝒫 (ω × 𝐴) ∈ V | 
| 5 | axdc3lem.2 | . . 3 ⊢ 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))} | |
| 6 | fssxp 6762 | . . . . . . . . 9 ⊢ (𝑠:suc 𝑛⟶𝐴 → 𝑠 ⊆ (suc 𝑛 × 𝐴)) | |
| 7 | peano2 7913 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
| 8 | omelon2 7901 | . . . . . . . . . . . 12 ⊢ (ω ∈ V → ω ∈ On) | |
| 9 | 1, 8 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ω ∈ On | 
| 10 | 9 | onelssi 6498 | . . . . . . . . . 10 ⊢ (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω) | 
| 11 | xpss1 5703 | . . . . . . . . . 10 ⊢ (suc 𝑛 ⊆ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴)) | |
| 12 | 7, 10, 11 | 3syl 18 | . . . . . . . . 9 ⊢ (𝑛 ∈ ω → (suc 𝑛 × 𝐴) ⊆ (ω × 𝐴)) | 
| 13 | 6, 12 | sylan9ss 3996 | . . . . . . . 8 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴)) | 
| 14 | velpw 4604 | . . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴)) | |
| 15 | 13, 14 | sylibr 234 | . . . . . . 7 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴)) | 
| 16 | 15 | ancoms 458 | . . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛⟶𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴)) | 
| 17 | 16 | 3ad2antr1 1188 | . . . . 5 ⊢ ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) | 
| 18 | 17 | rexlimiva 3146 | . . . 4 ⊢ (∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) | 
| 19 | 18 | abssi 4069 | . . 3 ⊢ {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))} ⊆ 𝒫 (ω × 𝐴) | 
| 20 | 5, 19 | eqsstri 4029 | . 2 ⊢ 𝑆 ⊆ 𝒫 (ω × 𝐴) | 
| 21 | 4, 20 | ssexi 5321 | 1 ⊢ 𝑆 ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 𝒫 cpw 4599 × cxp 5682 Oncon0 6383 suc csuc 6385 ⟶wf 6556 ‘cfv 6560 ωcom 7888 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-dc 10487 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-om 7889 df-1o 8507 | 
| This theorem is referenced by: axdc3lem2 10492 axdc3lem4 10494 | 
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