Theorem cshwsexa 14851

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 2772	. . . . 5 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 ↔ 𝑤 = (𝑊 cyclShift 𝑛))
2	1	rexbii 3112	. . . 4 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
3	2	abbii 2832	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)}
4		ovex 7433	. . . 4 ⊢ (0..^(♯‘𝑊)) ∈ V
5	4	abrexex 7947	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)} ∈ V
6	3, 5	eqeltri 2861	. 2 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
7		rabssab 4041	. 2 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ⊆ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
8	6, 7	ssexi 5283	1 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Syntax hints:   = wceq 1563   ∈ wcel 2145  {cab 2743  ∃wrex 3089  {crab 3417  Vcvv 3457  ‘cfv 6525  (class class class)co 7400  0cc0 11088  ..^cfzo 13673  ♯chash 14357  Word cword 14540   cyclShift ccsh 14815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-sn 4586  df-pr 4588  df-uni 4869  df-iota 6481  df-fv 6533  df-ov 7403

This theorem is referenced by: (None)