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Theorem wrdval 13537
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 4714 . . . 4 (𝑤 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆𝑚 (0..^𝑙)))
2 ovex 6910 . . . . . 6 (0..^𝑙) ∈ V
3 elmapg 8108 . . . . . 6 ((𝑆𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
42, 3mpan2 683 . . . . 5 (𝑆𝑉 → (𝑤 ∈ (𝑆𝑚 (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
54rexbidv 3233 . . . 4 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
61, 5syl5bb 275 . . 3 (𝑆𝑉 → (𝑤 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
76abbi2dv 2919 . 2 (𝑆𝑉 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
8 df-word 13535 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
97, 8syl6reqr 2852 1 (𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  {cab 2785  wrex 3090  Vcvv 3385   ciun 4710  wf 6097  (class class class)co 6878  𝑚 cmap 8095  0cc0 10224  0cn0 11580  ..^cfzo 12720  Word cword 13534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-word 13535
This theorem is referenced by:  wrdexg  13544
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