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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 9858 | . . . 4 ⊢ ω ∈ V | |
2 | axdc3lem.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | xpex 7456 | . . 3 ⊢ (ω × 𝐴) ∈ V |
4 | 3 | pwex 5246 | . 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 6508 | . . . . . . . . 9 ⊢ (𝑠:suc 𝑛⟶𝐴 → 𝑠 ⊆ (suc 𝑛 × 𝐴)) | |
7 | peano2 7582 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
8 | omelon2 7572 | . . . . . . . . . . . 12 ⊢ (ω ∈ V → ω ∈ On) | |
9 | 1, 8 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ω ∈ On |
10 | 9 | onelssi 6267 | . . . . . . . . . 10 ⊢ (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω) |
11 | xpss1 5538 | . . . . . . . . . 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 3928 | . . . . . . . 8 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴)) |
14 | velpw 4502 | . . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴)) | |
15 | 13, 14 | sylibr 237 | . . . . . . 7 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
16 | 15 | ancoms 462 | . . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛⟶𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
17 | 16 | 3ad2antr1 1185 | . . . . 5 ⊢ ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
18 | 17 | rexlimiva 3240 | . . . 4 ⊢ (∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
19 | 18 | abssi 3997 | . . 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 3949 | . 2 ⊢ 𝑆 ⊆ 𝒫 (ω × 𝐴) |
21 | 4, 20 | ssexi 5190 | 1 ⊢ 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 × cxp 5517 Oncon0 6159 suc csuc 6161 ⟶wf 6320 ‘cfv 6324 ωcom 7560 |
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-dc 9857 |
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-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-om 7561 df-1o 8085 |
This theorem is referenced by: axdc3lem2 9862 axdc3lem4 9864 |
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