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Theorem iswrd 13863
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 13862 . . 3 Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}
21eleq2i 2884 . 2 (𝑊 ∈ Word 𝑆𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆})
3 ovex 7172 . . . . 5 (0..^𝑙) ∈ V
4 fex 6970 . . . . 5 ((𝑊:(0..^𝑙)⟶𝑆 ∧ (0..^𝑙) ∈ V) → 𝑊 ∈ V)
53, 4mpan2 690 . . . 4 (𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
65rexlimivw 3244 . . 3 (∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆𝑊 ∈ V)
7 feq1 6472 . . . 4 (𝑤 = 𝑊 → (𝑤:(0..^𝑙)⟶𝑆𝑊:(0..^𝑙)⟶𝑆))
87rexbidv 3259 . . 3 (𝑤 = 𝑊 → (∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆))
96, 8elab3 3625 . 2 (𝑊 ∈ {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
102, 9bitri 278 1 (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  Vcvv 3444  wf 6324  (class class class)co 7139  0cc0 10530  0cn0 11889  ..^cfzo 13032  Word cword 13861
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-word 13862
This theorem is referenced by:  iswrdi  13865  wrdf  13866  cshword  14148  motcgrg  26342
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