Step |
Hyp |
Ref |
Expression |
0 |
|
crelexp |
⊢ ↑𝑟 |
1 |
|
vr |
⊢ 𝑟 |
2 |
|
cvv |
⊢ V |
3 |
|
vn |
⊢ 𝑛 |
4 |
|
cn0 |
⊢ ℕ0 |
5 |
3
|
cv |
⊢ 𝑛 |
6 |
|
cc0 |
⊢ 0 |
7 |
5 6
|
wceq |
⊢ 𝑛 = 0 |
8 |
|
cid |
⊢ I |
9 |
1
|
cv |
⊢ 𝑟 |
10 |
9
|
cdm |
⊢ dom 𝑟 |
11 |
9
|
crn |
⊢ ran 𝑟 |
12 |
10 11
|
cun |
⊢ ( dom 𝑟 ∪ ran 𝑟 ) |
13 |
8 12
|
cres |
⊢ ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) |
14 |
|
c1 |
⊢ 1 |
15 |
|
vx |
⊢ 𝑥 |
16 |
|
vy |
⊢ 𝑦 |
17 |
15
|
cv |
⊢ 𝑥 |
18 |
17 9
|
ccom |
⊢ ( 𝑥 ∘ 𝑟 ) |
19 |
15 16 2 2 18
|
cmpo |
⊢ ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) |
20 |
|
vz |
⊢ 𝑧 |
21 |
20 2 9
|
cmpt |
⊢ ( 𝑧 ∈ V ↦ 𝑟 ) |
22 |
19 21 14
|
cseq |
⊢ seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) |
23 |
5 22
|
cfv |
⊢ ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) |
24 |
7 13 23
|
cif |
⊢ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) |
25 |
1 3 2 4 24
|
cmpo |
⊢ ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) |
26 |
0 25
|
wceq |
⊢ ↑𝑟 = ( 𝑟 ∈ V , 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , ( I ↾ ( dom 𝑟 ∪ ran 𝑟 ) ) , ( seq 1 ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑥 ∘ 𝑟 ) ) , ( 𝑧 ∈ V ↦ 𝑟 ) ) ‘ 𝑛 ) ) ) |