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 e. u , y e. u |-> ( y ^m x ) ) |
17 |
9 16
|
cop |
|- <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( y ^m x ) ) >. |
18 |
|
cco |
|- comp |
19 |
4 18
|
cfv |
|- ( comp ` ndx ) |
20 |
|
vv |
|- v |
21 |
6 6
|
cxp |
|- ( u X. 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 o. f ) |
36 |
23 29 28 32 35
|
cmpo |
|- ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) |
37 |
20 22 21 6 36
|
cmpo |
|- ( v e. ( u X. u ) , z e. u |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) |
38 |
19 37
|
cop |
|- <. ( comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) >. |
39 |
7 17 38
|
ctp |
|- { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( y ^m x ) ) >. , <. ( comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) >. } |
40 |
1 2 39
|
cmpt |
|- ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( y ^m x ) ) >. , <. ( comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) >. } ) |
41 |
0 40
|
wceq |
|- SetCat = ( u e. _V |-> { <. ( Base ` ndx ) , u >. , <. ( Hom ` ndx ) , ( x e. u , y e. u |-> ( y ^m x ) ) >. , <. ( comp ` ndx ) , ( v e. ( u X. u ) , z e. u |-> ( g e. ( z ^m ( 2nd ` v ) ) , f e. ( ( 2nd ` v ) ^m ( 1st ` v ) ) |-> ( g o. f ) ) ) >. } ) |