![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cshwsexa | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cshwsexa | ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2738 | . . . . 5 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 ↔ 𝑤 = (𝑊 cyclShift 𝑛)) | |
2 | 1 | rexbii 3093 | . . . 4 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) |
3 | 2 | abbii 2801 | . . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)} |
4 | ovex 7426 | . . . 4 ⊢ (0..^(♯‘𝑊)) ∈ V | |
5 | 4 | abrexex 7931 | . . 3 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)} ∈ V |
6 | 3, 5 | eqeltri 2828 | . 2 ⊢ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V |
7 | rabssab 4079 | . 2 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ⊆ {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} | |
8 | 6, 7 | ssexi 5315 | 1 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 {cab 2708 ∃wrex 3069 {crab 3431 Vcvv 3473 ‘cfv 6532 (class class class)co 7393 0cc0 11092 ..^cfzo 13609 ♯chash 14272 Word cword 14446 cyclShift ccsh 14720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-sn 4623 df-pr 4625 df-uni 4902 df-iota 6484 df-fv 6540 df-ov 7396 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |