Step |
Hyp |
Ref |
Expression |
0 |
|
cmvr |
⊢ mVar |
1 |
|
vi |
⊢ 𝑖 |
2 |
|
cvv |
⊢ V |
3 |
|
vr |
⊢ 𝑟 |
4 |
|
vx |
⊢ 𝑥 |
5 |
1
|
cv |
⊢ 𝑖 |
6 |
|
vf |
⊢ 𝑓 |
7 |
|
vh |
⊢ ℎ |
8 |
|
cn0 |
⊢ ℕ0 |
9 |
|
cmap |
⊢ ↑m |
10 |
8 5 9
|
co |
⊢ ( ℕ0 ↑m 𝑖 ) |
11 |
7
|
cv |
⊢ ℎ |
12 |
11
|
ccnv |
⊢ ◡ ℎ |
13 |
|
cn |
⊢ ℕ |
14 |
12 13
|
cima |
⊢ ( ◡ ℎ “ ℕ ) |
15 |
|
cfn |
⊢ Fin |
16 |
14 15
|
wcel |
⊢ ( ◡ ℎ “ ℕ ) ∈ Fin |
17 |
16 7 10
|
crab |
⊢ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
18 |
6
|
cv |
⊢ 𝑓 |
19 |
|
vy |
⊢ 𝑦 |
20 |
19
|
cv |
⊢ 𝑦 |
21 |
4
|
cv |
⊢ 𝑥 |
22 |
20 21
|
wceq |
⊢ 𝑦 = 𝑥 |
23 |
|
c1 |
⊢ 1 |
24 |
|
cc0 |
⊢ 0 |
25 |
22 23 24
|
cif |
⊢ if ( 𝑦 = 𝑥 , 1 , 0 ) |
26 |
19 5 25
|
cmpt |
⊢ ( 𝑦 ∈ 𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) |
27 |
18 26
|
wceq |
⊢ 𝑓 = ( 𝑦 ∈ 𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) |
28 |
|
cur |
⊢ 1r |
29 |
3
|
cv |
⊢ 𝑟 |
30 |
29 28
|
cfv |
⊢ ( 1r ‘ 𝑟 ) |
31 |
|
c0g |
⊢ 0g |
32 |
29 31
|
cfv |
⊢ ( 0g ‘ 𝑟 ) |
33 |
27 30 32
|
cif |
⊢ if ( 𝑓 = ( 𝑦 ∈ 𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) |
34 |
6 17 33
|
cmpt |
⊢ ( 𝑓 ∈ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦 ∈ 𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) ) |
35 |
4 5 34
|
cmpt |
⊢ ( 𝑥 ∈ 𝑖 ↦ ( 𝑓 ∈ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦 ∈ 𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) ) ) |
36 |
1 3 2 2 35
|
cmpo |
⊢ ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑥 ∈ 𝑖 ↦ ( 𝑓 ∈ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦 ∈ 𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) ) ) ) |
37 |
0 36
|
wceq |
⊢ mVar = ( 𝑖 ∈ V , 𝑟 ∈ V ↦ ( 𝑥 ∈ 𝑖 ↦ ( 𝑓 ∈ { ℎ ∈ ( ℕ0 ↑m 𝑖 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } ↦ if ( 𝑓 = ( 𝑦 ∈ 𝑖 ↦ if ( 𝑦 = 𝑥 , 1 , 0 ) ) , ( 1r ‘ 𝑟 ) , ( 0g ‘ 𝑟 ) ) ) ) ) |