Theorem cshwsexa 14834

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 2768	. . . . 5 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 ↔ 𝑤 = (𝑊 cyclShift 𝑛))
2	1	rexbii 3108	. . . 4 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
3	2	abbii 2828	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)}
4		ovex 7425	. . . 4 ⊢ (0..^(♯‘𝑊)) ∈ V
5	4	abrexex 7939	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)} ∈ V
6	3, 5	eqeltri 2857	. 2 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
7		rabssab 4038	. 2 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ⊆ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
8	6, 7	ssexi 5277	1 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Syntax hints:   = wceq 1559   ∈ wcel 2141  {cab 2739  ∃wrex 3085  {crab 3413  Vcvv 3453  ‘cfv 6517  (class class class)co 7392  0cc0 11070  ..^cfzo 13656  ♯chash 14340  Word cword 14523   cyclShift ccsh 14798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6473  df-fv 6525  df-ov 7395

This theorem is referenced by: (None)