Step |
Hyp |
Ref |
Expression |
0 |
|
chof |
⊢ HomF |
1 |
|
vc |
⊢ 𝑐 |
2 |
|
ccat |
⊢ Cat |
3 |
|
chomf |
⊢ Homf |
4 |
1
|
cv |
⊢ 𝑐 |
5 |
4 3
|
cfv |
⊢ ( Homf ‘ 𝑐 ) |
6 |
|
cbs |
⊢ Base |
7 |
4 6
|
cfv |
⊢ ( Base ‘ 𝑐 ) |
8 |
|
vb |
⊢ 𝑏 |
9 |
|
vx |
⊢ 𝑥 |
10 |
8
|
cv |
⊢ 𝑏 |
11 |
10 10
|
cxp |
⊢ ( 𝑏 × 𝑏 ) |
12 |
|
vy |
⊢ 𝑦 |
13 |
|
vf |
⊢ 𝑓 |
14 |
|
c1st |
⊢ 1st |
15 |
12
|
cv |
⊢ 𝑦 |
16 |
15 14
|
cfv |
⊢ ( 1st ‘ 𝑦 ) |
17 |
|
chom |
⊢ Hom |
18 |
4 17
|
cfv |
⊢ ( Hom ‘ 𝑐 ) |
19 |
9
|
cv |
⊢ 𝑥 |
20 |
19 14
|
cfv |
⊢ ( 1st ‘ 𝑥 ) |
21 |
16 20 18
|
co |
⊢ ( ( 1st ‘ 𝑦 ) ( Hom ‘ 𝑐 ) ( 1st ‘ 𝑥 ) ) |
22 |
|
vg |
⊢ 𝑔 |
23 |
|
c2nd |
⊢ 2nd |
24 |
19 23
|
cfv |
⊢ ( 2nd ‘ 𝑥 ) |
25 |
15 23
|
cfv |
⊢ ( 2nd ‘ 𝑦 ) |
26 |
24 25 18
|
co |
⊢ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) |
27 |
|
vh |
⊢ ℎ |
28 |
19 18
|
cfv |
⊢ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) |
29 |
22
|
cv |
⊢ 𝑔 |
30 |
|
cco |
⊢ comp |
31 |
4 30
|
cfv |
⊢ ( comp ‘ 𝑐 ) |
32 |
19 25 31
|
co |
⊢ ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) |
33 |
27
|
cv |
⊢ ℎ |
34 |
29 33 32
|
co |
⊢ ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) |
35 |
16 20
|
cop |
⊢ 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 |
36 |
35 25 31
|
co |
⊢ ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) |
37 |
13
|
cv |
⊢ 𝑓 |
38 |
34 37 36
|
co |
⊢ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) |
39 |
27 28 38
|
cmpt |
⊢ ( ℎ ∈ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) |
40 |
13 22 21 26 39
|
cmpo |
⊢ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) ( Hom ‘ 𝑐 ) ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) |
41 |
9 12 11 11 40
|
cmpo |
⊢ ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ ( 𝑏 × 𝑏 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) ( Hom ‘ 𝑐 ) ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) |
42 |
8 7 41
|
csb |
⊢ ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ ( 𝑏 × 𝑏 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) ( Hom ‘ 𝑐 ) ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) |
43 |
5 42
|
cop |
⊢ 〈 ( Homf ‘ 𝑐 ) , ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ ( 𝑏 × 𝑏 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) ( Hom ‘ 𝑐 ) ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) 〉 |
44 |
1 2 43
|
cmpt |
⊢ ( 𝑐 ∈ Cat ↦ 〈 ( Homf ‘ 𝑐 ) , ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ ( 𝑏 × 𝑏 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) ( Hom ‘ 𝑐 ) ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) 〉 ) |
45 |
0 44
|
wceq |
⊢ HomF = ( 𝑐 ∈ Cat ↦ 〈 ( Homf ‘ 𝑐 ) , ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ( 𝑥 ∈ ( 𝑏 × 𝑏 ) , 𝑦 ∈ ( 𝑏 × 𝑏 ) ↦ ( 𝑓 ∈ ( ( 1st ‘ 𝑦 ) ( Hom ‘ 𝑐 ) ( 1st ‘ 𝑥 ) ) , 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ↦ ( ℎ ∈ ( ( Hom ‘ 𝑐 ) ‘ 𝑥 ) ↦ ( ( 𝑔 ( 𝑥 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) ℎ ) ( 〈 ( 1st ‘ 𝑦 ) , ( 1st ‘ 𝑥 ) 〉 ( comp ‘ 𝑐 ) ( 2nd ‘ 𝑦 ) ) 𝑓 ) ) ) ) 〉 ) |