Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iswrd Structured version   Visualization version   GIF version

Theorem iswrd 13162
 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 13154 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
21eleq2i 2680 . 2 (𝑊 ∈ Word 𝑆𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3 ovex 6577 . . . . 5 (0..^𝑙) ∈ V
4 fex 6394 . . . . 5 ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V)
53, 4mpan2 703 . . . 4 (𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
65rexlimivw 3011 . . 3 (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
7 feq1 5939 . . . 4 (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆𝑊:(0..^𝑙)⟶𝑆))
87rexbidv 3034 . . 3 (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆))
96, 8elab3 3327 . 2 (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
102, 9bitri 263 1 (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173  ⟶wf 5800  (class class class)co 6549  0cc0 9815  ℕ0cn0 11169  ..^cfzo 12334  Word cword 13146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-word 13154 This theorem is referenced by:  iswrdi  13164  wrdf  13165  cshword  13388  motcgrg  25239  cshword2  40300
 Copyright terms: Public domain W3C validator