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Theorem cshwsexa 14756
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 2738 . . . . 5 ((𝑊 cyclShift 𝑛) = 𝑤𝑤 = (𝑊 cyclShift 𝑛))
21rexbii 3093 . . . 4 (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
32abbii 2801 . . 3 {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)}
4 ovex 7426 . . . 4 (0..^(♯‘𝑊)) ∈ V
54abrexex 7931 . . 3 {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)} ∈ V
63, 5eqeltri 2828 . 2 {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
7 rabssab 4079 . 2 {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ⊆ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
86, 7ssexi 5315 1 {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  {cab 2708  wrex 3069  {crab 3431  Vcvv 3473  cfv 6532  (class class class)co 7393  0cc0 11092  ..^cfzo 13609  chash 14272  Word cword 14446   cyclShift ccsh 14720
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-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4623  df-pr 4625  df-uni 4902  df-iota 6484  df-fv 6540  df-ov 7396
This theorem is referenced by: (None)
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