Step |
Hyp |
Ref |
Expression |
1 |
|
diag2.l |
|- L = ( C DiagFunc D ) |
2 |
|
diag2.a |
|- A = ( Base ` C ) |
3 |
|
diag2.b |
|- B = ( Base ` D ) |
4 |
|
diag2.h |
|- H = ( Hom ` C ) |
5 |
|
diag2.c |
|- ( ph -> C e. Cat ) |
6 |
|
diag2.d |
|- ( ph -> D e. Cat ) |
7 |
|
diag2.x |
|- ( ph -> X e. A ) |
8 |
|
diag2.y |
|- ( ph -> Y e. A ) |
9 |
|
diag2.f |
|- ( ph -> F e. ( X H Y ) ) |
10 |
1 5 6
|
diagval |
|- ( ph -> L = ( <. C , D >. curryF ( C 1stF D ) ) ) |
11 |
10
|
fveq2d |
|- ( ph -> ( 2nd ` L ) = ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) ) |
12 |
11
|
oveqd |
|- ( ph -> ( X ( 2nd ` L ) Y ) = ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ) |
13 |
12
|
fveq1d |
|- ( ph -> ( ( X ( 2nd ` L ) Y ) ` F ) = ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F ) ) |
14 |
|
eqid |
|- ( <. C , D >. curryF ( C 1stF D ) ) = ( <. C , D >. curryF ( C 1stF D ) ) |
15 |
|
eqid |
|- ( C Xc. D ) = ( C Xc. D ) |
16 |
|
eqid |
|- ( C 1stF D ) = ( C 1stF D ) |
17 |
15 5 6 16
|
1stfcl |
|- ( ph -> ( C 1stF D ) e. ( ( C Xc. D ) Func C ) ) |
18 |
|
eqid |
|- ( Id ` D ) = ( Id ` D ) |
19 |
|
eqid |
|- ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F ) = ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F ) |
20 |
14 2 5 6 17 3 4 18 7 8 9 19
|
curf2 |
|- ( ph -> ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F ) = ( x e. B |-> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) ) ) |
21 |
15 2 3
|
xpcbas |
|- ( A X. B ) = ( Base ` ( C Xc. D ) ) |
22 |
|
eqid |
|- ( Hom ` ( C Xc. D ) ) = ( Hom ` ( C Xc. D ) ) |
23 |
5
|
adantr |
|- ( ( ph /\ x e. B ) -> C e. Cat ) |
24 |
6
|
adantr |
|- ( ( ph /\ x e. B ) -> D e. Cat ) |
25 |
|
opelxpi |
|- ( ( X e. A /\ x e. B ) -> <. X , x >. e. ( A X. B ) ) |
26 |
7 25
|
sylan |
|- ( ( ph /\ x e. B ) -> <. X , x >. e. ( A X. B ) ) |
27 |
|
opelxpi |
|- ( ( Y e. A /\ x e. B ) -> <. Y , x >. e. ( A X. B ) ) |
28 |
8 27
|
sylan |
|- ( ( ph /\ x e. B ) -> <. Y , x >. e. ( A X. B ) ) |
29 |
15 21 22 23 24 16 26 28
|
1stf2 |
|- ( ( ph /\ x e. B ) -> ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) = ( 1st |` ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) ) ) |
30 |
29
|
oveqd |
|- ( ( ph /\ x e. B ) -> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) = ( F ( 1st |` ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) ) ( ( Id ` D ) ` x ) ) ) |
31 |
|
df-ov |
|- ( F ( 1st |` ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) ) ( ( Id ` D ) ` x ) ) = ( ( 1st |` ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) ) ` <. F , ( ( Id ` D ) ` x ) >. ) |
32 |
9
|
adantr |
|- ( ( ph /\ x e. B ) -> F e. ( X H Y ) ) |
33 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
34 |
|
simpr |
|- ( ( ph /\ x e. B ) -> x e. B ) |
35 |
3 33 18 24 34
|
catidcl |
|- ( ( ph /\ x e. B ) -> ( ( Id ` D ) ` x ) e. ( x ( Hom ` D ) x ) ) |
36 |
32 35
|
opelxpd |
|- ( ( ph /\ x e. B ) -> <. F , ( ( Id ` D ) ` x ) >. e. ( ( X H Y ) X. ( x ( Hom ` D ) x ) ) ) |
37 |
7
|
adantr |
|- ( ( ph /\ x e. B ) -> X e. A ) |
38 |
8
|
adantr |
|- ( ( ph /\ x e. B ) -> Y e. A ) |
39 |
15 2 3 4 33 37 34 38 34 22
|
xpchom2 |
|- ( ( ph /\ x e. B ) -> ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) = ( ( X H Y ) X. ( x ( Hom ` D ) x ) ) ) |
40 |
36 39
|
eleqtrrd |
|- ( ( ph /\ x e. B ) -> <. F , ( ( Id ` D ) ` x ) >. e. ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) ) |
41 |
40
|
fvresd |
|- ( ( ph /\ x e. B ) -> ( ( 1st |` ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) ) ` <. F , ( ( Id ` D ) ` x ) >. ) = ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) ) |
42 |
31 41
|
eqtrid |
|- ( ( ph /\ x e. B ) -> ( F ( 1st |` ( <. X , x >. ( Hom ` ( C Xc. D ) ) <. Y , x >. ) ) ( ( Id ` D ) ` x ) ) = ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) ) |
43 |
|
op1stg |
|- ( ( F e. ( X H Y ) /\ ( ( Id ` D ) ` x ) e. ( x ( Hom ` D ) x ) ) -> ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) = F ) |
44 |
9 35 43
|
syl2an2r |
|- ( ( ph /\ x e. B ) -> ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) = F ) |
45 |
30 42 44
|
3eqtrd |
|- ( ( ph /\ x e. B ) -> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) = F ) |
46 |
45
|
mpteq2dva |
|- ( ph -> ( x e. B |-> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) ) = ( x e. B |-> F ) ) |
47 |
|
fconstmpt |
|- ( B X. { F } ) = ( x e. B |-> F ) |
48 |
46 47
|
eqtr4di |
|- ( ph -> ( x e. B |-> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) ) = ( B X. { F } ) ) |
49 |
13 20 48
|
3eqtrd |
|- ( ph -> ( ( X ( 2nd ` L ) Y ) ` F ) = ( B X. { F } ) ) |