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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 14474 | . 2 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙))) | |
2 | nn0ex 12485 | . . 3 ⊢ ℕ0 ∈ V | |
3 | ovexd 7447 | . . . 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 7954 | . . 3 ⊢ ((ℕ0 ∈ V ∧ ∀𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) | |
6 | 2, 4, 5 | sylancr 586 | . 2 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑙 ∈ ℕ0 (𝑆 ↑m (0..^𝑙)) ∈ V) |
7 | 1, 6 | eqeltrd 2832 | 1 ⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ∪ ciun 4997 (class class class)co 7412 ↑m cmap 8826 0cc0 11116 ℕ0cn0 12479 ..^cfzo 13634 Word cword 14471 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-1cn 11174 ax-addcl 11176 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-map 8828 df-nn 12220 df-n0 12480 df-word 14472 |
This theorem is referenced by: wrdexb 14482 wrdexi 14483 wrdnfi 14505 elovmpowrd 14515 elovmptnn0wrd 14516 wrd2f1tovbij 14918 frmdbas 18772 frmdplusg 18774 efgval 19630 frgp0 19673 frgpmhm 19678 vrgpf 19681 vrgpinv 19682 frgpupf 19686 frgpup1 19688 frgpup2 19689 frgpup3lem 19690 frgpnabllem1 19786 frgpnabllem2 19787 wksfval 29148 wksvOLD 29159 wwlks 29371 clwwlk 29518 tocycval 32552 sseqval 33700 upwlksfval 46824 |
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