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Theorem wrdval 14555
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 df-word 14553 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2 eliun 4995 . . . 4 (𝑤 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆m (0..^𝑙)))
3 ovex 7464 . . . . . 6 (0..^𝑙) ∈ V
4 elmapg 8879 . . . . . 6 ((𝑆𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
53, 4mpan2 691 . . . . 5 (𝑆𝑉 → (𝑤 ∈ (𝑆m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
65rexbidv 3179 . . . 4 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
72, 6bitrid 283 . . 3 (𝑆𝑉 → (𝑤 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
87eqabdv 2875 . 2 (𝑆𝑉 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
91, 8eqtr4id 2796 1 (𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480   ciun 4991  wf 6557  (class class class)co 7431  m cmap 8866  0cc0 11155  0cn0 12526  ..^cfzo 13694  Word cword 14552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-word 14553
This theorem is referenced by:  wrdexg  14562
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