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Theorem wrdval 13247
Description: Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
wrdval (𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)))
Distinct variable groups:   𝑆,𝑙   𝑉,𝑙

Proof of Theorem wrdval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eliun 4490 . . . 4 (𝑤 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆𝑚 (0..^𝑙)))
2 ovex 6632 . . . . . 6 (0..^𝑙) ∈ V
3 elmapg 7815 . . . . . 6 ((𝑆𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
42, 3mpan2 706 . . . . 5 (𝑆𝑉 → (𝑤 ∈ (𝑆𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
54rexbidv 3045 . . . 4 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
61, 5syl5bb 272 . . 3 (𝑆𝑉 → (𝑤 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
76abbi2dv 2739 . 2 (𝑆𝑉 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
8 df-word 13238 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
97, 8syl6reqr 2674 1 (𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186   ciun 4485  wf 5843  (class class class)co 6604  𝑚 cmap 7802  0cc0 9880  0cn0 11236  ..^cfzo 12406  Word cword 13230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-word 13238
This theorem is referenced by:  wrdexg  13254
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