Theorem cshwsexa 14186

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 3147	. . 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 3350	. . . . 5 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) ↔ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
3	2	bicomi 226	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))
4	3	abbii 2886	. . 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 3144	. . . 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 2886	. . 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 2848	. 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 2957	. . . 4 ⊢ {𝑛 ∣ 𝑛 ∈ (0..^(♯‘𝑊))} = (0..^(♯‘𝑊))
9	8	ovexi 7190	. . 3 ⊢ {𝑛 ∣ 𝑛 ∈ (0..^(♯‘𝑊))} ∈ V
10		tru 1541	. . . . 5 ⊢ ⊤
11	10, 10	pm3.2i 473	. . . 4 ⊢ (⊤ ∧ ⊤)
12		ovexd 7191	. . . . . 6 ⊢ (⊤ → (𝑊 cyclShift 𝑛) ∈ V)
13		eqtr3 2843	. . . . . . . . . . . . 13 ⊢ ((𝑤 = (𝑊 cyclShift 𝑛) ∧ 𝑦 = (𝑊 cyclShift 𝑛)) → 𝑤 = 𝑦)
14	13	ex 415	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
15	14	eqcoms 2829	. . . . . . . . . . 11 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 → (𝑦 = (𝑊 cyclShift 𝑛) → 𝑤 = 𝑦))
16	15	adantl 484	. . . . . . . . . 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 725	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) ∧ ⊤) → ((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
19	18	alrimiv 1928	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) ∧ ⊤) → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦))
20	19	ex 415	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 = (𝑊 cyclShift 𝑛)) → (⊤ → ∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦)))
21	12, 20	spcimedv 3594	. . . . 5 ⊢ (⊤ → (⊤ → ∃𝑦∀𝑤((𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤) → 𝑤 = 𝑦)))
22	21	imp 409	. . . 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 5200	. 2 ⊢ {𝑤 ∣ ∃𝑛(𝑛 ∈ (0..^(♯‘𝑊)) ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑊 cyclShift 𝑛) = 𝑤))} ∈ V
25	7, 24	eqeltri 2909	1 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537  ⊤wtru 1538  ∃wex 1780   ∈ wcel 2114  {cab 2799  ∃wrex 3139  {crab 3142  Vcvv 3494  ‘cfv 6355  (class class class)co 7156  0cc0 10537  ..^cfzo 13034  ♯chash 13691  Word cword 13862   cyclShift ccsh 14150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-nul 5210

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4568  df-pr 4570  df-uni 4839  df-iota 6314  df-fv 6363  df-ov 7159

This theorem is referenced by: (None)