Step |
Hyp |
Ref |
Expression |
0 |
|
cfunc |
⊢ Func |
1 |
|
vt |
⊢ 𝑡 |
2 |
|
ccat |
⊢ Cat |
3 |
|
vu |
⊢ 𝑢 |
4 |
|
vf |
⊢ 𝑓 |
5 |
|
vg |
⊢ 𝑔 |
6 |
|
cbs |
⊢ Base |
7 |
1
|
cv |
⊢ 𝑡 |
8 |
7 6
|
cfv |
⊢ ( Base ‘ 𝑡 ) |
9 |
|
vb |
⊢ 𝑏 |
10 |
4
|
cv |
⊢ 𝑓 |
11 |
9
|
cv |
⊢ 𝑏 |
12 |
3
|
cv |
⊢ 𝑢 |
13 |
12 6
|
cfv |
⊢ ( Base ‘ 𝑢 ) |
14 |
11 13 10
|
wf |
⊢ 𝑓 : 𝑏 ⟶ ( Base ‘ 𝑢 ) |
15 |
5
|
cv |
⊢ 𝑔 |
16 |
|
vz |
⊢ 𝑧 |
17 |
11 11
|
cxp |
⊢ ( 𝑏 × 𝑏 ) |
18 |
|
c1st |
⊢ 1st |
19 |
16
|
cv |
⊢ 𝑧 |
20 |
19 18
|
cfv |
⊢ ( 1st ‘ 𝑧 ) |
21 |
20 10
|
cfv |
⊢ ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) |
22 |
|
chom |
⊢ Hom |
23 |
12 22
|
cfv |
⊢ ( Hom ‘ 𝑢 ) |
24 |
|
c2nd |
⊢ 2nd |
25 |
19 24
|
cfv |
⊢ ( 2nd ‘ 𝑧 ) |
26 |
25 10
|
cfv |
⊢ ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) |
27 |
21 26 23
|
co |
⊢ ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) |
28 |
|
cmap |
⊢ ↑m |
29 |
7 22
|
cfv |
⊢ ( Hom ‘ 𝑡 ) |
30 |
19 29
|
cfv |
⊢ ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) |
31 |
27 30 28
|
co |
⊢ ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) |
32 |
16 17 31
|
cixp |
⊢ X 𝑧 ∈ ( 𝑏 × 𝑏 ) ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) |
33 |
15 32
|
wcel |
⊢ 𝑔 ∈ X 𝑧 ∈ ( 𝑏 × 𝑏 ) ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) |
34 |
|
vx |
⊢ 𝑥 |
35 |
34
|
cv |
⊢ 𝑥 |
36 |
35 35 15
|
co |
⊢ ( 𝑥 𝑔 𝑥 ) |
37 |
|
ccid |
⊢ Id |
38 |
7 37
|
cfv |
⊢ ( Id ‘ 𝑡 ) |
39 |
35 38
|
cfv |
⊢ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) |
40 |
39 36
|
cfv |
⊢ ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) |
41 |
12 37
|
cfv |
⊢ ( Id ‘ 𝑢 ) |
42 |
35 10
|
cfv |
⊢ ( 𝑓 ‘ 𝑥 ) |
43 |
42 41
|
cfv |
⊢ ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) |
44 |
40 43
|
wceq |
⊢ ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) |
45 |
|
vy |
⊢ 𝑦 |
46 |
|
vm |
⊢ 𝑚 |
47 |
45
|
cv |
⊢ 𝑦 |
48 |
35 47 29
|
co |
⊢ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) |
49 |
|
vn |
⊢ 𝑛 |
50 |
47 19 29
|
co |
⊢ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) |
51 |
35 19 15
|
co |
⊢ ( 𝑥 𝑔 𝑧 ) |
52 |
49
|
cv |
⊢ 𝑛 |
53 |
35 47
|
cop |
⊢ 〈 𝑥 , 𝑦 〉 |
54 |
|
cco |
⊢ comp |
55 |
7 54
|
cfv |
⊢ ( comp ‘ 𝑡 ) |
56 |
53 19 55
|
co |
⊢ ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) |
57 |
46
|
cv |
⊢ 𝑚 |
58 |
52 57 56
|
co |
⊢ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) |
59 |
58 51
|
cfv |
⊢ ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) |
60 |
47 19 15
|
co |
⊢ ( 𝑦 𝑔 𝑧 ) |
61 |
52 60
|
cfv |
⊢ ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) |
62 |
47 10
|
cfv |
⊢ ( 𝑓 ‘ 𝑦 ) |
63 |
42 62
|
cop |
⊢ 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 |
64 |
12 54
|
cfv |
⊢ ( comp ‘ 𝑢 ) |
65 |
19 10
|
cfv |
⊢ ( 𝑓 ‘ 𝑧 ) |
66 |
63 65 64
|
co |
⊢ ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) |
67 |
35 47 15
|
co |
⊢ ( 𝑥 𝑔 𝑦 ) |
68 |
57 67
|
cfv |
⊢ ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) |
69 |
61 68 66
|
co |
⊢ ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) |
70 |
59 69
|
wceq |
⊢ ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) |
71 |
70 49 50
|
wral |
⊢ ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) |
72 |
71 46 48
|
wral |
⊢ ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) |
73 |
72 16 11
|
wral |
⊢ ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) |
74 |
73 45 11
|
wral |
⊢ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) |
75 |
44 74
|
wa |
⊢ ( ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) ) |
76 |
75 34 11
|
wral |
⊢ ∀ 𝑥 ∈ 𝑏 ( ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) ) |
77 |
14 33 76
|
w3a |
⊢ ( 𝑓 : 𝑏 ⟶ ( Base ‘ 𝑢 ) ∧ 𝑔 ∈ X 𝑧 ∈ ( 𝑏 × 𝑏 ) ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝑏 ( ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) ) ) |
78 |
77 9 8
|
wsbc |
⊢ [ ( Base ‘ 𝑡 ) / 𝑏 ] ( 𝑓 : 𝑏 ⟶ ( Base ‘ 𝑢 ) ∧ 𝑔 ∈ X 𝑧 ∈ ( 𝑏 × 𝑏 ) ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝑏 ( ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) ) ) |
79 |
78 4 5
|
copab |
⊢ { 〈 𝑓 , 𝑔 〉 ∣ [ ( Base ‘ 𝑡 ) / 𝑏 ] ( 𝑓 : 𝑏 ⟶ ( Base ‘ 𝑢 ) ∧ 𝑔 ∈ X 𝑧 ∈ ( 𝑏 × 𝑏 ) ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝑏 ( ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) ) ) } |
80 |
1 3 2 2 79
|
cmpo |
⊢ ( 𝑡 ∈ Cat , 𝑢 ∈ Cat ↦ { 〈 𝑓 , 𝑔 〉 ∣ [ ( Base ‘ 𝑡 ) / 𝑏 ] ( 𝑓 : 𝑏 ⟶ ( Base ‘ 𝑢 ) ∧ 𝑔 ∈ X 𝑧 ∈ ( 𝑏 × 𝑏 ) ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝑏 ( ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) ) ) } ) |
81 |
0 80
|
wceq |
⊢ Func = ( 𝑡 ∈ Cat , 𝑢 ∈ Cat ↦ { 〈 𝑓 , 𝑔 〉 ∣ [ ( Base ‘ 𝑡 ) / 𝑏 ] ( 𝑓 : 𝑏 ⟶ ( Base ‘ 𝑢 ) ∧ 𝑔 ∈ X 𝑧 ∈ ( 𝑏 × 𝑏 ) ( ( ( 𝑓 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝑢 ) ( 𝑓 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝑡 ) ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝑏 ( ( ( 𝑥 𝑔 𝑥 ) ‘ ( ( Id ‘ 𝑡 ) ‘ 𝑥 ) ) = ( ( Id ‘ 𝑢 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝑡 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝑡 ) 𝑧 ) ( ( 𝑥 𝑔 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑡 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝑔 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) 〉 ( comp ‘ 𝑢 ) ( 𝑓 ‘ 𝑧 ) ) ( ( 𝑥 𝑔 𝑦 ) ‘ 𝑚 ) ) ) ) } ) |