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Theorem iswrd 14445
Description: Property of being a word over a set with an existential quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)
Assertion
Ref Expression
iswrd (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
Distinct variable groups:   𝑆,𝑙   𝑊,𝑙

Proof of Theorem iswrd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-word 14444 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
21eleq2i 2824 . 2 (𝑊 ∈ Word 𝑆𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3 ovex 7423 . . . . 5 (0..^𝑙) ∈ V
4 fex 7209 . . . . 5 ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V)
53, 4mpan2 689 . . . 4 (𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
65rexlimivw 3150 . . 3 (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
7 feq1 6682 . . . 4 (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆𝑊:(0..^𝑙)⟶𝑆))
87rexbidv 3177 . . 3 (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆))
96, 8elab3 3669 . 2 (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
102, 9bitri 274 1 (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  {cab 2708  wrex 3069  Vcvv 3470  wf 6525  (class class class)co 7390  0cc0 11089  0cn0 12451  ..^cfzo 13606  Word cword 14443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-ov 7393  df-word 14444
This theorem is referenced by:  iswrdi  14447  wrdf  14448  cshword  14720  motcgrg  27655
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