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Theorem cshwsexa 14859
Description: The class of (different!) words resulting by cyclically shifting something (not necessarily a word) is a set. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) (Proof shortened by SN, 15-Jan-2025.)
Assertion
Ref Expression
cshwsexa {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
Distinct variable group:   𝑛,𝑊,𝑤
Allowed substitution hints:   𝑉(𝑤,𝑛)

Proof of Theorem cshwsexa
StepHypRef Expression
1 eqcom 2742 . . . . 5 ((𝑊 cyclShift 𝑛) = 𝑤𝑤 = (𝑊 cyclShift 𝑛))
21rexbii 3092 . . . 4 (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
32abbii 2807 . . 3 {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)}
4 ovex 7464 . . . 4 (0..^(♯‘𝑊)) ∈ V
54abrexex 7986 . . 3 {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)} ∈ V
63, 5eqeltri 2835 . 2 {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
7 rabssab 4095 . 2 {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ⊆ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
86, 7ssexi 5328 1 {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433  Vcvv 3478  cfv 6563  (class class class)co 7431  0cc0 11153  ..^cfzo 13691  chash 14366  Word cword 14549   cyclShift ccsh 14823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by: (None)
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