Step |
Hyp |
Ref |
Expression |
0 |
|
c1stf |
|- 1stF |
1 |
|
vr |
|- r |
2 |
|
ccat |
|- Cat |
3 |
|
vs |
|- s |
4 |
|
cbs |
|- Base |
5 |
1
|
cv |
|- r |
6 |
5 4
|
cfv |
|- ( Base ` r ) |
7 |
3
|
cv |
|- s |
8 |
7 4
|
cfv |
|- ( Base ` s ) |
9 |
6 8
|
cxp |
|- ( ( Base ` r ) X. ( Base ` s ) ) |
10 |
|
vb |
|- b |
11 |
|
c1st |
|- 1st |
12 |
10
|
cv |
|- b |
13 |
11 12
|
cres |
|- ( 1st |` b ) |
14 |
|
vx |
|- x |
15 |
|
vy |
|- y |
16 |
14
|
cv |
|- x |
17 |
|
chom |
|- Hom |
18 |
|
cxpc |
|- Xc. |
19 |
5 7 18
|
co |
|- ( r Xc. s ) |
20 |
19 17
|
cfv |
|- ( Hom ` ( r Xc. s ) ) |
21 |
15
|
cv |
|- y |
22 |
16 21 20
|
co |
|- ( x ( Hom ` ( r Xc. s ) ) y ) |
23 |
11 22
|
cres |
|- ( 1st |` ( x ( Hom ` ( r Xc. s ) ) y ) ) |
24 |
14 15 12 12 23
|
cmpo |
|- ( x e. b , y e. b |-> ( 1st |` ( x ( Hom ` ( r Xc. s ) ) y ) ) ) |
25 |
13 24
|
cop |
|- <. ( 1st |` b ) , ( x e. b , y e. b |-> ( 1st |` ( x ( Hom ` ( r Xc. s ) ) y ) ) ) >. |
26 |
10 9 25
|
csb |
|- [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ <. ( 1st |` b ) , ( x e. b , y e. b |-> ( 1st |` ( x ( Hom ` ( r Xc. s ) ) y ) ) ) >. |
27 |
1 3 2 2 26
|
cmpo |
|- ( r e. Cat , s e. Cat |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ <. ( 1st |` b ) , ( x e. b , y e. b |-> ( 1st |` ( x ( Hom ` ( r Xc. s ) ) y ) ) ) >. ) |
28 |
0 27
|
wceq |
|- 1stF = ( r e. Cat , s e. Cat |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ <. ( 1st |` b ) , ( x e. b , y e. b |-> ( 1st |` ( x ( Hom ` ( r Xc. s ) ) y ) ) ) >. ) |