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Theorem iswrd 13485
Description: Property of being a word over a set with a 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 13477 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
21eleq2i 2823 . 2 (𝑊 ∈ Word 𝑆𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3 ovex 6833 . . . . 5 (0..^𝑙) ∈ V
4 fex 6645 . . . . 5 ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V)
53, 4mpan2 709 . . . 4 (𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
65rexlimivw 3159 . . 3 (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
7 feq1 6179 . . . 4 (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆𝑊:(0..^𝑙)⟶𝑆))
87rexbidv 3182 . . 3 (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆))
96, 8elab3 3490 . 2 (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
102, 9bitri 264 1 (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1624  wcel 2131  {cab 2738  wrex 3043  Vcvv 3332  wf 6037  (class class class)co 6805  0cc0 10120  0cn0 11476  ..^cfzo 12651  Word cword 13469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-word 13477
This theorem is referenced by:  iswrdi  13487  wrdf  13488  cshword  13729  motcgrg  25630  cshword2  41939
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