Theorem cshwsexa 14842

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

Step	Hyp	Ref	Expression
1		eqcom 2742	. . . . 5 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 ↔ 𝑤 = (𝑊 cyclShift 𝑛))
2	1	rexbii 3083	. . . 4 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
3	2	abbii 2802	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)}
4		ovex 7438	. . . 4 ⊢ (0..^(♯‘𝑊)) ∈ V
5	4	abrexex 7961	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)} ∈ V
6	3, 5	eqeltri 2830	. 2 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
7		rabssab 4060	. 2 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ⊆ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
8	6, 7	ssexi 5292	1 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Syntax hints:   = wceq 1540   ∈ wcel 2108  {cab 2713  ∃wrex 3060  {crab 3415  Vcvv 3459  ‘cfv 6531  (class class class)co 7405  0cc0 11129  ..^cfzo 13671  ♯chash 14348  Word cword 14531   cyclShift ccsh 14806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-sn 4602  df-pr 4604  df-uni 4884  df-iota 6484  df-fv 6539  df-ov 7408

This theorem is referenced by: (None)