Theorem cshwsexa 14027

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.)

Assertion

Ref	Expression
cshwsexa	⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Distinct variable groups:   𝑛,𝑉   𝑛,𝑊,𝑤

Allowed substitution hint:   𝑉(𝑤)

Proof of Theorem cshwsexa

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3114	. . 3 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)}
2		r19.42v 3311	. . . . 5 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) ↔ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
3	2	bicomi 225	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))
4	3	abbii 2861	. . 3 ⊢ {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤)}
5		df-rex 3111	. . . 4 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤)))
6	5	abbii 2861	. . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤)} = {𝑤 ∣ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))}
7	1, 4, 6	3eqtri 2823	. 2 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))}
8		abid2 2926	. . . 4 ⊢ {𝑛 ∣ 𝑛 ∈ (0..^(♯‘𝑊))} = (0..^(♯‘𝑊))
9	8	ovexi 7054	. . 3 ⊢ {𝑛 ∣ 𝑛 ∈ (0..^(♯‘𝑊))} ∈ V
10		tru 1526	. . . . 5 ⊢ ⊤
11	10, 10	pm3.2i 471	. . . 4 ⊢ (⊤ ∧ ⊤)
12		ovexd 7055	. . . . . 6 ⊢ (⊤ → (𝑊 cyclShift 𝑛) ∈ V)
13		eqtr3 2818	. . . . . . . . . . . . 13 ⊢ ((𝑤 = (𝑊 cyclShift 𝑛) ∧ 𝑦 = (𝑊 cyclShift 𝑛)) → 𝑤 = 𝑦)
14	13	ex 413	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
15	14	eqcoms 2803	. . . . . . . . . . 11 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
16	15	adantl 482	. . . . . . . . . 10 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
17	16	com12 32	. . . . . . . . 9 ⊢ (𝑦 = (𝑊 cyclShift 𝑛) → ((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
18	17	ad2antlr 723	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) ∧ ⊤) → ((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
19	18	alrimiv 1905	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) ∧ ⊤) → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
20	19	ex 413	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) → (⊤ → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦)))
21	12, 20	spcimedv 3537	. . . . 5 ⊢ (⊤ → (⊤ → ∃𝑦∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦)))
22	21	imp 407	. . . 4 ⊢ ((⊤ ∧ ⊤) → ∃𝑦∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
23	11, 22	mp1i 13	. . 3 ⊢ (𝑛 ∈ (0..^(♯‘𝑊)) → ∃𝑦∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
24	9, 23	zfrep4 5096	. 2 ⊢ {𝑤 ∣ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))} ∈ V
25	7, 24	eqeltri 2879	1 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1520   = wceq 1522  ⊤wtru 1523  ∃wex 1761   ∈ wcel 2081  {cab 2775  ∃wrex 3106  {crab 3109  Vcvv 3437  ‘cfv 6230  (class class class)co 7021  0cc0 10388  ..^cfzo 12888  ♯chash 13545  Word cword 13712   cyclShift ccsh 13991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-rep 5086  ax-nul 5106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-sn 4477  df-pr 4479  df-uni 4750  df-iota 6194  df-fv 6238  df-ov 7024

This theorem is referenced by: (None)