Step |
Hyp |
Ref |
Expression |
0 |
|
cidfu |
⊢ idfunc |
1 |
|
vt |
⊢ 𝑡 |
2 |
|
ccat |
⊢ Cat |
3 |
|
cbs |
⊢ Base |
4 |
1
|
cv |
⊢ 𝑡 |
5 |
4 3
|
cfv |
⊢ ( Base ‘ 𝑡 ) |
6 |
|
vb |
⊢ 𝑏 |
7 |
|
cid |
⊢ I |
8 |
6
|
cv |
⊢ 𝑏 |
9 |
7 8
|
cres |
⊢ ( I ↾ 𝑏 ) |
10 |
|
vz |
⊢ 𝑧 |
11 |
8 8
|
cxp |
⊢ ( 𝑏 × 𝑏 ) |
12 |
|
chom |
⊢ Hom |
13 |
4 12
|
cfv |
⊢ ( Hom ‘ 𝑡 ) |
14 |
10
|
cv |
⊢ 𝑧 |
15 |
14 13
|
cfv |
⊢ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) |
16 |
7 15
|
cres |
⊢ ( I ↾ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) |
17 |
10 11 16
|
cmpt |
⊢ ( 𝑧 ∈ ( 𝑏 × 𝑏 ) ↦ ( I ↾ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ) |
18 |
9 17
|
cop |
⊢ 〈 ( I ↾ 𝑏 ) , ( 𝑧 ∈ ( 𝑏 × 𝑏 ) ↦ ( I ↾ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ) 〉 |
19 |
6 5 18
|
csb |
⊢ ⦋ ( Base ‘ 𝑡 ) / 𝑏 ⦌ 〈 ( I ↾ 𝑏 ) , ( 𝑧 ∈ ( 𝑏 × 𝑏 ) ↦ ( I ↾ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ) 〉 |
20 |
1 2 19
|
cmpt |
⊢ ( 𝑡 ∈ Cat ↦ ⦋ ( Base ‘ 𝑡 ) / 𝑏 ⦌ 〈 ( I ↾ 𝑏 ) , ( 𝑧 ∈ ( 𝑏 × 𝑏 ) ↦ ( I ↾ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ) 〉 ) |
21 |
0 20
|
wceq |
⊢ idfunc = ( 𝑡 ∈ Cat ↦ ⦋ ( Base ‘ 𝑡 ) / 𝑏 ⦌ 〈 ( I ↾ 𝑏 ) , ( 𝑧 ∈ ( 𝑏 × 𝑏 ) ↦ ( I ↾ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ) 〉 ) |