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Theorem wrdval 13852
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 4914 . . . 4 (𝑤 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆m (0..^𝑙)))
2 ovex 7178 . . . . . 6 (0..^𝑙) ∈ V
3 elmapg 8408 . . . . . 6 ((𝑆𝑉 ∧ (0..^𝑙) ∈ V) → (𝑤 ∈ (𝑆m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
42, 3mpan2 687 . . . . 5 (𝑆𝑉 → (𝑤 ∈ (𝑆m (0..^𝑙)) ↔ 𝑤:(0..^𝑙)⟶𝑆))
54rexbidv 3294 . . . 4 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
61, 5syl5bb 284 . . 3 (𝑆𝑉 → (𝑤 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
76abbi2dv 2947 . 2 (𝑆𝑉 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
8 df-word 13850 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
97, 8syl6reqr 2872 1 (𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492   ciun 4910  wf 6344  (class class class)co 7145  m cmap 8395  0cc0 10525  0cn0 11885  ..^cfzo 13021  Word cword 13849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-word 13850
This theorem is referenced by:  wrdexg  13859  wrdexgOLD  13860
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