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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.) (Proof shortened by JJ, 18-Nov-2022.) | 
| Ref | Expression | 
|---|---|
| wrdexg | ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | wrdval 14556 | . 2 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙))) | |
| 2 | nn0ex 12534 | . . 3 ⊢ ℕ0 ∈ V | |
| 3 | ovexd 7467 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑙 ∈ ℕ0) → (𝑆 ↑m (0..^𝑙)) ∈ V) | |
| 4 | 3 | ralrimiva 3145 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∀𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) | 
| 5 | iunexg 7989 | . . 3 ⊢ ((ℕ0 ∈ V ∧ ∀𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) | |
| 6 | 2, 4, 5 | sylancr 587 | . 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) | 
| 7 | 1, 6 | eqeltrd 2840 | 1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∪ ciun 4990 (class class class)co 7432 ↑m cmap 8867 0cc0 11156 ℕ0cn0 12528 ..^cfzo 13695 Word cword 14553 | 
| 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-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-1cn 11214 ax-addcl 11216 | 
| 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-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 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-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 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-res 5696 df-ima 5697 df-pred 6320 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-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-map 8869 df-nn 12268 df-n0 12529 df-word 14554 | 
| This theorem is referenced by: wrdexb 14564 wrdexi 14565 wrdnfi 14587 elovmpowrd 14597 elovmptnn0wrd 14598 wrd2f1tovbij 15000 frmdbas 18866 frmdplusg 18868 efgval 19736 frgp0 19779 frgpmhm 19784 vrgpf 19787 vrgpinv 19788 frgpupf 19792 frgpup1 19794 frgpup2 19795 frgpup3lem 19796 frgpnabllem1 19892 frgpnabllem2 19893 wksfval 29628 wksvOLD 29639 wwlks 29856 clwwlk 30003 gsumwrd2dccat 33071 tocycval 33129 elrgspnlem1 33247 elrgspnlem2 33248 elrgspnlem3 33249 elrgspnlem4 33250 elrgspn 33251 elrgspnsubrunlem1 33252 elrgspnsubrunlem2 33253 elrgspnsubrun 33254 sseqval 34391 upwlksfval 48056 | 
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