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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 10444 | . . . 4 ⊢ ω ∈ V | |
2 | axdc3lem.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | xpex 7742 | . . 3 ⊢ (ω × 𝐴) ∈ V |
4 | 3 | pwex 5377 | . 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 6744 | . . . . . . . . 9 ⊢ (𝑠:suc 𝑛⟶𝐴 → 𝑠 ⊆ (suc 𝑛 × 𝐴)) | |
7 | peano2 7883 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
8 | omelon2 7870 | . . . . . . . . . . . 12 ⊢ (ω ∈ V → ω ∈ On) | |
9 | 1, 8 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ω ∈ On |
10 | 9 | onelssi 6478 | . . . . . . . . . 10 ⊢ (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω) |
11 | xpss1 5694 | . . . . . . . . . 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 3994 | . . . . . . . 8 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴)) |
14 | velpw 4606 | . . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴)) | |
15 | 13, 14 | sylibr 233 | . . . . . . 7 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
16 | 15 | ancoms 457 | . . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛⟶𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
17 | 16 | 3ad2antr1 1186 | . . . . 5 ⊢ ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
18 | 17 | rexlimiva 3145 | . . . 4 ⊢ (∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
19 | 18 | abssi 4066 | . . 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 4015 | . 2 ⊢ 𝑆 ⊆ 𝒫 (ω × 𝐴) |
21 | 4, 20 | ssexi 5321 | 1 ⊢ 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 {cab 2707 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 × cxp 5673 Oncon0 6363 suc csuc 6365 ⟶wf 6538 ‘cfv 6542 ωcom 7857 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-dc 10443 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-om 7858 df-1o 8468 |
This theorem is referenced by: axdc3lem2 10448 axdc3lem4 10450 |
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