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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 9869 | . . . 4 ⊢ ω ∈ V | |
2 | axdc3lem.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | xpex 7476 | . . 3 ⊢ (ω × 𝐴) ∈ V |
4 | 3 | pwex 5281 | . 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 6534 | . . . . . . . . 9 ⊢ (𝑠:suc 𝑛⟶𝐴 → 𝑠 ⊆ (suc 𝑛 × 𝐴)) | |
7 | peano2 7602 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω) | |
8 | omelon2 7592 | . . . . . . . . . . . 12 ⊢ (ω ∈ V → ω ∈ On) | |
9 | 1, 8 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ω ∈ On |
10 | 9 | onelssi 6299 | . . . . . . . . . 10 ⊢ (suc 𝑛 ∈ ω → suc 𝑛 ⊆ ω) |
11 | xpss1 5574 | . . . . . . . . . 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 3980 | . . . . . . . 8 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ⊆ (ω × 𝐴)) |
14 | velpw 4544 | . . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 (ω × 𝐴) ↔ 𝑠 ⊆ (ω × 𝐴)) | |
15 | 13, 14 | sylibr 236 | . . . . . . 7 ⊢ ((𝑠:suc 𝑛⟶𝐴 ∧ 𝑛 ∈ ω) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
16 | 15 | ancoms 461 | . . . . . 6 ⊢ ((𝑛 ∈ ω ∧ 𝑠:suc 𝑛⟶𝐴) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
17 | 16 | 3ad2antr1 1184 | . . . . 5 ⊢ ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘)))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
18 | 17 | rexlimiva 3281 | . . . 4 ⊢ (∃𝑛 ∈ ω (𝑠:suc 𝑛⟶𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘 ∈ 𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠‘𝑘))) → 𝑠 ∈ 𝒫 (ω × 𝐴)) |
19 | 18 | abssi 4046 | . . 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 4001 | . 2 ⊢ 𝑆 ⊆ 𝒫 (ω × 𝐴) |
21 | 4, 20 | ssexi 5226 | 1 ⊢ 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 × cxp 5553 Oncon0 6191 suc csuc 6193 ⟶wf 6351 ‘cfv 6355 ωcom 7580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-dc 9868 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-om 7581 df-1o 8102 |
This theorem is referenced by: axdc3lem2 9873 axdc3lem4 9875 |
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