Detailed syntax breakdown of Definition df-ack
Step | Hyp | Ref
| Expression |
1 | | cack 45422 |
. 2
class
Ack |
2 | | vf |
. . . 4
setvar 𝑓 |
3 | | vj |
. . . 4
setvar 𝑗 |
4 | | cvv 3407 |
. . . 4
class
V |
5 | | vn |
. . . . 5
setvar 𝑛 |
6 | | cn0 11919 |
. . . . 5
class
ℕ0 |
7 | | c1 10561 |
. . . . . 6
class
1 |
8 | 5 | cv 1538 |
. . . . . . . 8
class 𝑛 |
9 | | caddc 10563 |
. . . . . . . 8
class
+ |
10 | 8, 7, 9 | co 7143 |
. . . . . . 7
class (𝑛 + 1) |
11 | 2 | cv 1538 |
. . . . . . . 8
class 𝑓 |
12 | | citco 45421 |
. . . . . . . 8
class
IterComp |
13 | 11, 12 | cfv 6328 |
. . . . . . 7
class
(IterComp‘𝑓) |
14 | 10, 13 | cfv 6328 |
. . . . . 6
class
((IterComp‘𝑓)‘(𝑛 + 1)) |
15 | 7, 14 | cfv 6328 |
. . . . 5
class
(((IterComp‘𝑓)‘(𝑛 + 1))‘1) |
16 | 5, 6, 15 | cmpt 5105 |
. . . 4
class (𝑛 ∈ ℕ0
↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1)) |
17 | 2, 3, 4, 4, 16 | cmpo 7145 |
. . 3
class (𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0 ↦
(((IterComp‘𝑓)‘(𝑛 + 1))‘1))) |
18 | | vi |
. . . 4
setvar 𝑖 |
19 | 18 | cv 1538 |
. . . . . 6
class 𝑖 |
20 | | cc0 10560 |
. . . . . 6
class
0 |
21 | 19, 20 | wceq 1539 |
. . . . 5
wff 𝑖 = 0 |
22 | 5, 6, 10 | cmpt 5105 |
. . . . 5
class (𝑛 ∈ ℕ0
↦ (𝑛 +
1)) |
23 | 21, 22, 19 | cif 4413 |
. . . 4
class if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖) |
24 | 18, 6, 23 | cmpt 5105 |
. . 3
class (𝑖 ∈ ℕ0
↦ if(𝑖 = 0, (𝑛 ∈ ℕ0
↦ (𝑛 + 1)), 𝑖)) |
25 | 17, 24, 20 | cseq 13403 |
. 2
class
seq0((𝑓 ∈ V,
𝑗 ∈ V ↦ (𝑛 ∈ ℕ0
↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))) |
26 | 1, 25 | wceq 1539 |
1
wff Ack =
seq0((𝑓 ∈ V, 𝑗 ∈ V ↦ (𝑛 ∈ ℕ0
↦ (((IterComp‘𝑓)‘(𝑛 + 1))‘1))), (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, (𝑛 ∈ ℕ0 ↦ (𝑛 + 1)), 𝑖))) |