Step |
Hyp |
Ref |
Expression |
0 |
|
csetc |
⊢ SetCat |
1 |
|
vu |
⊢ 𝑢 |
2 |
|
cvv |
⊢ V |
3 |
|
cbs |
⊢ Base |
4 |
|
cnx |
⊢ ndx |
5 |
4 3
|
cfv |
⊢ ( Base ‘ ndx ) |
6 |
1
|
cv |
⊢ 𝑢 |
7 |
5 6
|
cop |
⊢ 〈 ( Base ‘ ndx ) , 𝑢 〉 |
8 |
|
chom |
⊢ Hom |
9 |
4 8
|
cfv |
⊢ ( Hom ‘ ndx ) |
10 |
|
vx |
⊢ 𝑥 |
11 |
|
vy |
⊢ 𝑦 |
12 |
11
|
cv |
⊢ 𝑦 |
13 |
|
cmap |
⊢ ↑m |
14 |
10
|
cv |
⊢ 𝑥 |
15 |
12 14 13
|
co |
⊢ ( 𝑦 ↑m 𝑥 ) |
16 |
10 11 6 6 15
|
cmpo |
⊢ ( 𝑥 ∈ 𝑢 , 𝑦 ∈ 𝑢 ↦ ( 𝑦 ↑m 𝑥 ) ) |
17 |
9 16
|
cop |
⊢ 〈 ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑢 , 𝑦 ∈ 𝑢 ↦ ( 𝑦 ↑m 𝑥 ) ) 〉 |
18 |
|
cco |
⊢ comp |
19 |
4 18
|
cfv |
⊢ ( comp ‘ ndx ) |
20 |
|
vv |
⊢ 𝑣 |
21 |
6 6
|
cxp |
⊢ ( 𝑢 × 𝑢 ) |
22 |
|
vz |
⊢ 𝑧 |
23 |
|
vg |
⊢ 𝑔 |
24 |
22
|
cv |
⊢ 𝑧 |
25 |
|
c2nd |
⊢ 2nd |
26 |
20
|
cv |
⊢ 𝑣 |
27 |
26 25
|
cfv |
⊢ ( 2nd ‘ 𝑣 ) |
28 |
24 27 13
|
co |
⊢ ( 𝑧 ↑m ( 2nd ‘ 𝑣 ) ) |
29 |
|
vf |
⊢ 𝑓 |
30 |
|
c1st |
⊢ 1st |
31 |
26 30
|
cfv |
⊢ ( 1st ‘ 𝑣 ) |
32 |
27 31 13
|
co |
⊢ ( ( 2nd ‘ 𝑣 ) ↑m ( 1st ‘ 𝑣 ) ) |
33 |
23
|
cv |
⊢ 𝑔 |
34 |
29
|
cv |
⊢ 𝑓 |
35 |
33 34
|
ccom |
⊢ ( 𝑔 ∘ 𝑓 ) |
36 |
23 29 28 32 35
|
cmpo |
⊢ ( 𝑔 ∈ ( 𝑧 ↑m ( 2nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) ↑m ( 1st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) |
37 |
20 22 21 6 36
|
cmpo |
⊢ ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧 ∈ 𝑢 ↦ ( 𝑔 ∈ ( 𝑧 ↑m ( 2nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) ↑m ( 1st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) |
38 |
19 37
|
cop |
⊢ 〈 ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧 ∈ 𝑢 ↦ ( 𝑔 ∈ ( 𝑧 ↑m ( 2nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) ↑m ( 1st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) 〉 |
39 |
7 17 38
|
ctp |
⊢ { 〈 ( Base ‘ ndx ) , 𝑢 〉 , 〈 ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑢 , 𝑦 ∈ 𝑢 ↦ ( 𝑦 ↑m 𝑥 ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧 ∈ 𝑢 ↦ ( 𝑔 ∈ ( 𝑧 ↑m ( 2nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) ↑m ( 1st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) 〉 } |
40 |
1 2 39
|
cmpt |
⊢ ( 𝑢 ∈ V ↦ { 〈 ( Base ‘ ndx ) , 𝑢 〉 , 〈 ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑢 , 𝑦 ∈ 𝑢 ↦ ( 𝑦 ↑m 𝑥 ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧 ∈ 𝑢 ↦ ( 𝑔 ∈ ( 𝑧 ↑m ( 2nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) ↑m ( 1st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) 〉 } ) |
41 |
0 40
|
wceq |
⊢ SetCat = ( 𝑢 ∈ V ↦ { 〈 ( Base ‘ ndx ) , 𝑢 〉 , 〈 ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑢 , 𝑦 ∈ 𝑢 ↦ ( 𝑦 ↑m 𝑥 ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑢 × 𝑢 ) , 𝑧 ∈ 𝑢 ↦ ( 𝑔 ∈ ( 𝑧 ↑m ( 2nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2nd ‘ 𝑣 ) ↑m ( 1st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) 〉 } ) |