Step |
Hyp |
Ref |
Expression |
0 |
|
cprf |
|- pairF |
1 |
|
vf |
|- f |
2 |
|
cvv |
|- _V |
3 |
|
vg |
|- g |
4 |
|
c1st |
|- 1st |
5 |
1
|
cv |
|- f |
6 |
5 4
|
cfv |
|- ( 1st ` f ) |
7 |
6
|
cdm |
|- dom ( 1st ` f ) |
8 |
|
vb |
|- b |
9 |
|
vx |
|- x |
10 |
8
|
cv |
|- b |
11 |
9
|
cv |
|- x |
12 |
11 6
|
cfv |
|- ( ( 1st ` f ) ` x ) |
13 |
3
|
cv |
|- g |
14 |
13 4
|
cfv |
|- ( 1st ` g ) |
15 |
11 14
|
cfv |
|- ( ( 1st ` g ) ` x ) |
16 |
12 15
|
cop |
|- <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. |
17 |
9 10 16
|
cmpt |
|- ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) |
18 |
|
vy |
|- y |
19 |
|
vh |
|- h |
20 |
|
c2nd |
|- 2nd |
21 |
5 20
|
cfv |
|- ( 2nd ` f ) |
22 |
18
|
cv |
|- y |
23 |
11 22 21
|
co |
|- ( x ( 2nd ` f ) y ) |
24 |
23
|
cdm |
|- dom ( x ( 2nd ` f ) y ) |
25 |
19
|
cv |
|- h |
26 |
25 23
|
cfv |
|- ( ( x ( 2nd ` f ) y ) ` h ) |
27 |
13 20
|
cfv |
|- ( 2nd ` g ) |
28 |
11 22 27
|
co |
|- ( x ( 2nd ` g ) y ) |
29 |
25 28
|
cfv |
|- ( ( x ( 2nd ` g ) y ) ` h ) |
30 |
26 29
|
cop |
|- <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. |
31 |
19 24 30
|
cmpt |
|- ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) |
32 |
9 18 10 10 31
|
cmpo |
|- ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) |
33 |
17 32
|
cop |
|- <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. |
34 |
8 7 33
|
csb |
|- [_ dom ( 1st ` f ) / b ]_ <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. |
35 |
1 3 2 2 34
|
cmpo |
|- ( f e. _V , g e. _V |-> [_ dom ( 1st ` f ) / b ]_ <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. ) |
36 |
0 35
|
wceq |
|- pairF = ( f e. _V , g e. _V |-> [_ dom ( 1st ` f ) / b ]_ <. ( x e. b |-> <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. ) , ( x e. b , y e. b |-> ( h e. dom ( x ( 2nd ` f ) y ) |-> <. ( ( x ( 2nd ` f ) y ) ` h ) , ( ( x ( 2nd ` g ) y ) ` h ) >. ) ) >. ) |