Step |
Hyp |
Ref |
Expression |
0 |
|
ctocyc |
⊢ toCyc |
1 |
|
vd |
⊢ 𝑑 |
2 |
|
cvv |
⊢ V |
3 |
|
vw |
⊢ 𝑤 |
4 |
|
vu |
⊢ 𝑢 |
5 |
1
|
cv |
⊢ 𝑑 |
6 |
5
|
cword |
⊢ Word 𝑑 |
7 |
4
|
cv |
⊢ 𝑢 |
8 |
7
|
cdm |
⊢ dom 𝑢 |
9 |
8 5 7
|
wf1 |
⊢ 𝑢 : dom 𝑢 –1-1→ 𝑑 |
10 |
9 4 6
|
crab |
⊢ { 𝑢 ∈ Word 𝑑 ∣ 𝑢 : dom 𝑢 –1-1→ 𝑑 } |
11 |
|
cid |
⊢ I |
12 |
3
|
cv |
⊢ 𝑤 |
13 |
12
|
crn |
⊢ ran 𝑤 |
14 |
5 13
|
cdif |
⊢ ( 𝑑 ∖ ran 𝑤 ) |
15 |
11 14
|
cres |
⊢ ( I ↾ ( 𝑑 ∖ ran 𝑤 ) ) |
16 |
|
ccsh |
⊢ cyclShift |
17 |
|
c1 |
⊢ 1 |
18 |
12 17 16
|
co |
⊢ ( 𝑤 cyclShift 1 ) |
19 |
12
|
ccnv |
⊢ ◡ 𝑤 |
20 |
18 19
|
ccom |
⊢ ( ( 𝑤 cyclShift 1 ) ∘ ◡ 𝑤 ) |
21 |
15 20
|
cun |
⊢ ( ( I ↾ ( 𝑑 ∖ ran 𝑤 ) ) ∪ ( ( 𝑤 cyclShift 1 ) ∘ ◡ 𝑤 ) ) |
22 |
3 10 21
|
cmpt |
⊢ ( 𝑤 ∈ { 𝑢 ∈ Word 𝑑 ∣ 𝑢 : dom 𝑢 –1-1→ 𝑑 } ↦ ( ( I ↾ ( 𝑑 ∖ ran 𝑤 ) ) ∪ ( ( 𝑤 cyclShift 1 ) ∘ ◡ 𝑤 ) ) ) |
23 |
1 2 22
|
cmpt |
⊢ ( 𝑑 ∈ V ↦ ( 𝑤 ∈ { 𝑢 ∈ Word 𝑑 ∣ 𝑢 : dom 𝑢 –1-1→ 𝑑 } ↦ ( ( I ↾ ( 𝑑 ∖ ran 𝑤 ) ) ∪ ( ( 𝑤 cyclShift 1 ) ∘ ◡ 𝑤 ) ) ) ) |
24 |
0 23
|
wceq |
⊢ toCyc = ( 𝑑 ∈ V ↦ ( 𝑤 ∈ { 𝑢 ∈ Word 𝑑 ∣ 𝑢 : dom 𝑢 –1-1→ 𝑑 } ↦ ( ( I ↾ ( 𝑑 ∖ ran 𝑤 ) ) ∪ ( ( 𝑤 cyclShift 1 ) ∘ ◡ 𝑤 ) ) ) ) |