Step |
Hyp |
Ref |
Expression |
0 |
|
ccatc |
|- CatCat |
1 |
|
vu |
|- u |
2 |
|
cvv |
|- _V |
3 |
1
|
cv |
|- u |
4 |
|
ccat |
|- Cat |
5 |
3 4
|
cin |
|- ( u i^i Cat ) |
6 |
|
vb |
|- b |
7 |
|
cbs |
|- Base |
8 |
|
cnx |
|- ndx |
9 |
8 7
|
cfv |
|- ( Base ` ndx ) |
10 |
6
|
cv |
|- b |
11 |
9 10
|
cop |
|- <. ( Base ` ndx ) , b >. |
12 |
|
chom |
|- Hom |
13 |
8 12
|
cfv |
|- ( Hom ` ndx ) |
14 |
|
vx |
|- x |
15 |
|
vy |
|- y |
16 |
14
|
cv |
|- x |
17 |
|
cfunc |
|- Func |
18 |
15
|
cv |
|- y |
19 |
16 18 17
|
co |
|- ( x Func y ) |
20 |
14 15 10 10 19
|
cmpo |
|- ( x e. b , y e. b |-> ( x Func y ) ) |
21 |
13 20
|
cop |
|- <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. |
22 |
|
cco |
|- comp |
23 |
8 22
|
cfv |
|- ( comp ` ndx ) |
24 |
|
vv |
|- v |
25 |
10 10
|
cxp |
|- ( b X. b ) |
26 |
|
vz |
|- z |
27 |
|
vg |
|- g |
28 |
|
c2nd |
|- 2nd |
29 |
24
|
cv |
|- v |
30 |
29 28
|
cfv |
|- ( 2nd ` v ) |
31 |
26
|
cv |
|- z |
32 |
30 31 17
|
co |
|- ( ( 2nd ` v ) Func z ) |
33 |
|
vf |
|- f |
34 |
29 17
|
cfv |
|- ( Func ` v ) |
35 |
27
|
cv |
|- g |
36 |
|
ccofu |
|- o.func |
37 |
33
|
cv |
|- f |
38 |
35 37 36
|
co |
|- ( g o.func f ) |
39 |
27 33 32 34 38
|
cmpo |
|- ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) |
40 |
24 26 25 10 39
|
cmpo |
|- ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) |
41 |
23 40
|
cop |
|- <. ( comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. |
42 |
11 21 41
|
ctp |
|- { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. ( comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } |
43 |
6 5 42
|
csb |
|- [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. ( comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } |
44 |
1 2 43
|
cmpt |
|- ( u e. _V |-> [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. ( comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } ) |
45 |
0 44
|
wceq |
|- CatCat = ( u e. _V |-> [_ ( u i^i Cat ) / b ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , ( x e. b , y e. b |-> ( x Func y ) ) >. , <. ( comp ` ndx ) , ( v e. ( b X. b ) , z e. b |-> ( g e. ( ( 2nd ` v ) Func z ) , f e. ( Func ` v ) |-> ( g o.func f ) ) ) >. } ) |