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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 13858 | . 2 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙))) | |
2 | nn0ex 11897 | . . 3 ⊢ ℕ0 ∈ V | |
3 | ovexd 7185 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑙 ∈ ℕ0) → (𝑆 ↑m (0..^𝑙)) ∈ V) | |
4 | 3 | ralrimiva 3182 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∀𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) |
5 | iunexg 7658 | . . 3 ⊢ ((ℕ0 ∈ V ∧ ∀𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) | |
6 | 2, 4, 5 | sylancr 589 | . 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) |
7 | 1, 6 | eqeltrd 2913 | 1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∪ ciun 4911 (class class class)co 7150 ↑m cmap 8400 0cc0 10531 ℕ0cn0 11891 ..^cfzo 13027 Word cword 13855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-map 8402 df-nn 11633 df-n0 11892 df-word 13856 |
This theorem is referenced by: wrdexb 13867 wrdexi 13868 wrdnfi 13893 wrdnfiOLD 13894 elovmpowrd 13904 elovmptnn0wrd 13905 wrd2f1tovbij 14318 frmdbas 18011 frmdplusg 18013 efgval 18837 frgp0 18880 frgpmhm 18885 vrgpf 18888 vrgpinv 18889 frgpupf 18893 frgpup1 18895 frgpup2 18896 frgpup3lem 18897 frgpnabllem1 18987 frgpnabllem2 18988 wksfval 27385 wksv 27395 wwlks 27607 clwwlk 27755 tocycval 30745 sseqval 31641 upwlksfval 44004 |
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