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Theorem wrdval 13857
 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 4920 . . . 4 (𝑤 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) ↔ ∃𝑙 ∈ ℕ0 𝑤 ∈ (𝑆m (0..^𝑙)))
2 ovex 7184 . . . . . 6 (0..^𝑙) ∈ V
3 elmapg 8412 . . . . . 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 3301 . . . 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 2954 . 2 (𝑆𝑉 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)) = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
8 df-word 13855 . 2 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
97, 8syl6reqr 2879 1 (𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆m (0..^𝑙)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1530   ∈ wcel 2106  {cab 2802  ∃wrex 3143  Vcvv 3499  ∪ ciun 4916  ⟶wf 6347  (class class class)co 7151   ↑m cmap 8399  0cc0 10529  ℕ0cn0 11889  ..^cfzo 13026  Word cword 13854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401  df-word 13855 This theorem is referenced by:  wrdexg  13864  wrdexgOLD  13865
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