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Mirrors > Home > MPE Home > Th. List > wrdexg | Structured version Visualization version GIF version |
Description: The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
wrdexg | ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdval 13340 | . 2 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙))) | |
2 | mapsspw 7935 | . . . . . 6 ⊢ (𝑆 ↑𝑚 (0..^𝑙)) ⊆ 𝒫 ((0..^𝑙) × 𝑆) | |
3 | elfzoelz 12509 | . . . . . . . . 9 ⊢ (𝑠 ∈ (0..^𝑙) → 𝑠 ∈ ℤ) | |
4 | 3 | ssriv 3640 | . . . . . . . 8 ⊢ (0..^𝑙) ⊆ ℤ |
5 | xpss1 5161 | . . . . . . . 8 ⊢ ((0..^𝑙) ⊆ ℤ → ((0..^𝑙) × 𝑆) ⊆ (ℤ × 𝑆)) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ ((0..^𝑙) × 𝑆) ⊆ (ℤ × 𝑆) |
7 | sspwb 4947 | . . . . . . 7 ⊢ (((0..^𝑙) × 𝑆) ⊆ (ℤ × 𝑆) ↔ 𝒫 ((0..^𝑙) × 𝑆) ⊆ 𝒫 (ℤ × 𝑆)) | |
8 | 6, 7 | mpbi 220 | . . . . . 6 ⊢ 𝒫 ((0..^𝑙) × 𝑆) ⊆ 𝒫 (ℤ × 𝑆) |
9 | 2, 8 | sstri 3645 | . . . . 5 ⊢ (𝑆 ↑𝑚 (0..^𝑙)) ⊆ 𝒫 (ℤ × 𝑆) |
10 | 9 | rgenw 2953 | . . . 4 ⊢ ∀𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ⊆ 𝒫 (ℤ × 𝑆) |
11 | iunss 4593 | . . . 4 ⊢ (∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ⊆ 𝒫 (ℤ × 𝑆) ↔ ∀𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ⊆ 𝒫 (ℤ × 𝑆)) | |
12 | 10, 11 | mpbir 221 | . . 3 ⊢ ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ⊆ 𝒫 (ℤ × 𝑆) |
13 | zex 11424 | . . . . 5 ⊢ ℤ ∈ V | |
14 | xpexg 7002 | . . . . 5 ⊢ ((ℤ ∈ V ∧ 𝑆 ∈ 𝑉) → (ℤ × 𝑆) ∈ V) | |
15 | 13, 14 | mpan 706 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (ℤ × 𝑆) ∈ V) |
16 | pwexg 4880 | . . . 4 ⊢ ((ℤ × 𝑆) ∈ V → 𝒫 (ℤ × 𝑆) ∈ V) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝒫 (ℤ × 𝑆) ∈ V) |
18 | ssexg 4837 | . . 3 ⊢ ((∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ⊆ 𝒫 (ℤ × 𝑆) ∧ 𝒫 (ℤ × 𝑆) ∈ V) → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ∈ V) | |
19 | 12, 17, 18 | sylancr 696 | . 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑𝑚 (0..^𝑙)) ∈ V) |
20 | 1, 19 | eqeltrd 2730 | 1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 ⊆ wss 3607 𝒫 cpw 4191 ∪ ciun 4552 × cxp 5141 (class class class)co 6690 ↑𝑚 cmap 7899 0cc0 9974 ℕ0cn0 11330 ℤcz 11415 ..^cfzo 12504 Word cword 13323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-map 7901 df-pm 7902 df-neg 10307 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-word 13331 |
This theorem is referenced by: wrdexb 13348 wrdexi 13349 wrdnfi 13370 elovmpt2wrd 13380 elovmptnn0wrd 13381 wrd2f1tovbij 13749 frmdbas 17436 frmdplusg 17438 vrmdfval 17440 efgval 18176 frgp0 18219 frgpmhm 18224 vrgpf 18227 vrgpinv 18228 frgpupf 18232 frgpup1 18234 frgpup2 18235 frgpup3lem 18236 frgpnabllem1 18322 frgpnabllem2 18323 ablfaclem1 18530 israg 25637 wksfval 26561 wksv 26571 wwlks 26783 clwwlk 26951 sseqval 30578 upwlksfval 42041 |
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