Step |
Hyp |
Ref |
Expression |
0 |
|
ccur- |
⊢ curry_ |
1 |
|
vx |
⊢ 𝑥 |
2 |
|
cvv |
⊢ V |
3 |
|
vy |
⊢ 𝑦 |
4 |
|
vz |
⊢ 𝑧 |
5 |
|
vf |
⊢ 𝑓 |
6 |
1
|
cv |
⊢ 𝑥 |
7 |
3
|
cv |
⊢ 𝑦 |
8 |
6 7
|
cxp |
⊢ ( 𝑥 × 𝑦 ) |
9 |
|
csethom |
⊢ Set⟶ |
10 |
4
|
cv |
⊢ 𝑧 |
11 |
8 10 9
|
co |
⊢ ( ( 𝑥 × 𝑦 ) Set⟶ 𝑧 ) |
12 |
|
va |
⊢ 𝑎 |
13 |
|
vb |
⊢ 𝑏 |
14 |
5
|
cv |
⊢ 𝑓 |
15 |
12
|
cv |
⊢ 𝑎 |
16 |
13
|
cv |
⊢ 𝑏 |
17 |
15 16
|
cop |
⊢ 〈 𝑎 , 𝑏 〉 |
18 |
17 14
|
cfv |
⊢ ( 𝑓 ‘ 〈 𝑎 , 𝑏 〉 ) |
19 |
13 7 18
|
cmpt |
⊢ ( 𝑏 ∈ 𝑦 ↦ ( 𝑓 ‘ 〈 𝑎 , 𝑏 〉 ) ) |
20 |
12 6 19
|
cmpt |
⊢ ( 𝑎 ∈ 𝑥 ↦ ( 𝑏 ∈ 𝑦 ↦ ( 𝑓 ‘ 〈 𝑎 , 𝑏 〉 ) ) ) |
21 |
5 11 20
|
cmpt |
⊢ ( 𝑓 ∈ ( ( 𝑥 × 𝑦 ) Set⟶ 𝑧 ) ↦ ( 𝑎 ∈ 𝑥 ↦ ( 𝑏 ∈ 𝑦 ↦ ( 𝑓 ‘ 〈 𝑎 , 𝑏 〉 ) ) ) ) |
22 |
1 3 4 2 2 2 21
|
cmpt3 |
⊢ ( 𝑥 ∈ V , 𝑦 ∈ V , 𝑧 ∈ V ↦ ( 𝑓 ∈ ( ( 𝑥 × 𝑦 ) Set⟶ 𝑧 ) ↦ ( 𝑎 ∈ 𝑥 ↦ ( 𝑏 ∈ 𝑦 ↦ ( 𝑓 ‘ 〈 𝑎 , 𝑏 〉 ) ) ) ) ) |
23 |
0 22
|
wceq |
⊢ curry_ = ( 𝑥 ∈ V , 𝑦 ∈ V , 𝑧 ∈ V ↦ ( 𝑓 ∈ ( ( 𝑥 × 𝑦 ) Set⟶ 𝑧 ) ↦ ( 𝑎 ∈ 𝑥 ↦ ( 𝑏 ∈ 𝑦 ↦ ( 𝑓 ‘ 〈 𝑎 , 𝑏 〉 ) ) ) ) ) |