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Theorem wrdval 13970
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 13968 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
2 eliun 4895 . . . 4 (𝑤 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆m (0..^𝑙)))
3 ovex 7215 . . . . . 6 (0..^𝑙) ∈ V
4 elmapg 8462 . . . . . 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 3208 . . . 4 (𝑆𝑉 → (∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
72, 6syl5bb 286 . . 3 (𝑆𝑉 → (𝑤 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆))
87abbi2dv 2870 . 2 (𝑆𝑉 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
91, 8eqtr4id 2793 1 (𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1542  wcel 2114  {cab 2717  wrex 3055  Vcvv 3400   ciun 4891  wf 6345  (class class class)co 7182  m cmap 8449  0cc0 10627  0cn0 11988  ..^cfzo 13136  Word cword 13967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-map 8451  df-word 13968
This theorem is referenced by:  wrdexg  13977
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