Step |
Hyp |
Ref |
Expression |
0 |
|
cinito |
⊢ InitO |
1 |
|
vc |
⊢ 𝑐 |
2 |
|
ccat |
⊢ Cat |
3 |
|
va |
⊢ 𝑎 |
4 |
|
cbs |
⊢ Base |
5 |
1
|
cv |
⊢ 𝑐 |
6 |
5 4
|
cfv |
⊢ ( Base ‘ 𝑐 ) |
7 |
|
vb |
⊢ 𝑏 |
8 |
|
vh |
⊢ ℎ |
9 |
8
|
cv |
⊢ ℎ |
10 |
3
|
cv |
⊢ 𝑎 |
11 |
|
chom |
⊢ Hom |
12 |
5 11
|
cfv |
⊢ ( Hom ‘ 𝑐 ) |
13 |
7
|
cv |
⊢ 𝑏 |
14 |
10 13 12
|
co |
⊢ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) |
15 |
9 14
|
wcel |
⊢ ℎ ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) |
16 |
15 8
|
weu |
⊢ ∃! ℎ ℎ ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) |
17 |
16 7 6
|
wral |
⊢ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ℎ ℎ ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) |
18 |
17 3 6
|
crab |
⊢ { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ℎ ℎ ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) } |
19 |
1 2 18
|
cmpt |
⊢ ( 𝑐 ∈ Cat ↦ { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ℎ ℎ ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) } ) |
20 |
0 19
|
wceq |
⊢ InitO = ( 𝑐 ∈ Cat ↦ { 𝑎 ∈ ( Base ‘ 𝑐 ) ∣ ∀ 𝑏 ∈ ( Base ‘ 𝑐 ) ∃! ℎ ℎ ∈ ( 𝑎 ( Hom ‘ 𝑐 ) 𝑏 ) } ) |