Step |
Hyp |
Ref |
Expression |
1 |
|
diagval.l |
|- L = ( C DiagFunc D ) |
2 |
|
diagval.c |
|- ( ph -> C e. Cat ) |
3 |
|
diagval.d |
|- ( ph -> D e. Cat ) |
4 |
|
diag11.a |
|- A = ( Base ` C ) |
5 |
|
diag11.c |
|- ( ph -> X e. A ) |
6 |
|
diag11.k |
|- K = ( ( 1st ` L ) ` X ) |
7 |
|
diag11.b |
|- B = ( Base ` D ) |
8 |
|
diag11.y |
|- ( ph -> Y e. B ) |
9 |
|
diag12.j |
|- J = ( Hom ` D ) |
10 |
|
diag12.i |
|- .1. = ( Id ` C ) |
11 |
|
diag12.z |
|- ( ph -> Z e. B ) |
12 |
|
diag12.f |
|- ( ph -> F e. ( Y J Z ) ) |
13 |
1 2 3
|
diagval |
|- ( ph -> L = ( <. C , D >. curryF ( C 1stF D ) ) ) |
14 |
13
|
fveq2d |
|- ( ph -> ( 1st ` L ) = ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ) |
15 |
14
|
fveq1d |
|- ( ph -> ( ( 1st ` L ) ` X ) = ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) ) |
16 |
6 15
|
eqtrid |
|- ( ph -> K = ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) ) |
17 |
16
|
fveq2d |
|- ( ph -> ( 2nd ` K ) = ( 2nd ` ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) ) ) |
18 |
17
|
oveqd |
|- ( ph -> ( Y ( 2nd ` K ) Z ) = ( Y ( 2nd ` ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) ) Z ) ) |
19 |
18
|
fveq1d |
|- ( ph -> ( ( Y ( 2nd ` K ) Z ) ` F ) = ( ( Y ( 2nd ` ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) ) Z ) ` F ) ) |
20 |
|
eqid |
|- ( <. C , D >. curryF ( C 1stF D ) ) = ( <. C , D >. curryF ( C 1stF D ) ) |
21 |
|
eqid |
|- ( C Xc. D ) = ( C Xc. D ) |
22 |
|
eqid |
|- ( C 1stF D ) = ( C 1stF D ) |
23 |
21 2 3 22
|
1stfcl |
|- ( ph -> ( C 1stF D ) e. ( ( C Xc. D ) Func C ) ) |
24 |
|
eqid |
|- ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) = ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) |
25 |
20 4 2 3 23 7 5 24 8 9 10 11 12
|
curf12 |
|- ( ph -> ( ( Y ( 2nd ` ( ( 1st ` ( <. C , D >. curryF ( C 1stF D ) ) ) ` X ) ) Z ) ` F ) = ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` ( C 1stF D ) ) <. X , Z >. ) F ) ) |
26 |
|
df-ov |
|- ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` ( C 1stF D ) ) <. X , Z >. ) F ) = ( ( <. X , Y >. ( 2nd ` ( C 1stF D ) ) <. X , Z >. ) ` <. ( .1. ` X ) , F >. ) |
27 |
21 4 7
|
xpcbas |
|- ( A X. B ) = ( Base ` ( C Xc. D ) ) |
28 |
|
eqid |
|- ( Hom ` ( C Xc. D ) ) = ( Hom ` ( C Xc. D ) ) |
29 |
5 8
|
opelxpd |
|- ( ph -> <. X , Y >. e. ( A X. B ) ) |
30 |
5 11
|
opelxpd |
|- ( ph -> <. X , Z >. e. ( A X. B ) ) |
31 |
21 27 28 2 3 22 29 30
|
1stf2 |
|- ( ph -> ( <. X , Y >. ( 2nd ` ( C 1stF D ) ) <. X , Z >. ) = ( 1st |` ( <. X , Y >. ( Hom ` ( C Xc. D ) ) <. X , Z >. ) ) ) |
32 |
31
|
fveq1d |
|- ( ph -> ( ( <. X , Y >. ( 2nd ` ( C 1stF D ) ) <. X , Z >. ) ` <. ( .1. ` X ) , F >. ) = ( ( 1st |` ( <. X , Y >. ( Hom ` ( C Xc. D ) ) <. X , Z >. ) ) ` <. ( .1. ` X ) , F >. ) ) |
33 |
26 32
|
eqtrid |
|- ( ph -> ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` ( C 1stF D ) ) <. X , Z >. ) F ) = ( ( 1st |` ( <. X , Y >. ( Hom ` ( C Xc. D ) ) <. X , Z >. ) ) ` <. ( .1. ` X ) , F >. ) ) |
34 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
35 |
4 34 10 2 5
|
catidcl |
|- ( ph -> ( .1. ` X ) e. ( X ( Hom ` C ) X ) ) |
36 |
35 12
|
opelxpd |
|- ( ph -> <. ( .1. ` X ) , F >. e. ( ( X ( Hom ` C ) X ) X. ( Y J Z ) ) ) |
37 |
21 4 7 34 9 5 8 5 11 28
|
xpchom2 |
|- ( ph -> ( <. X , Y >. ( Hom ` ( C Xc. D ) ) <. X , Z >. ) = ( ( X ( Hom ` C ) X ) X. ( Y J Z ) ) ) |
38 |
36 37
|
eleqtrrd |
|- ( ph -> <. ( .1. ` X ) , F >. e. ( <. X , Y >. ( Hom ` ( C Xc. D ) ) <. X , Z >. ) ) |
39 |
38
|
fvresd |
|- ( ph -> ( ( 1st |` ( <. X , Y >. ( Hom ` ( C Xc. D ) ) <. X , Z >. ) ) ` <. ( .1. ` X ) , F >. ) = ( 1st ` <. ( .1. ` X ) , F >. ) ) |
40 |
|
op1stg |
|- ( ( ( .1. ` X ) e. ( X ( Hom ` C ) X ) /\ F e. ( Y J Z ) ) -> ( 1st ` <. ( .1. ` X ) , F >. ) = ( .1. ` X ) ) |
41 |
35 12 40
|
syl2anc |
|- ( ph -> ( 1st ` <. ( .1. ` X ) , F >. ) = ( .1. ` X ) ) |
42 |
33 39 41
|
3eqtrd |
|- ( ph -> ( ( .1. ` X ) ( <. X , Y >. ( 2nd ` ( C 1stF D ) ) <. X , Z >. ) F ) = ( .1. ` X ) ) |
43 |
19 25 42
|
3eqtrd |
|- ( ph -> ( ( Y ( 2nd ` K ) Z ) ` F ) = ( .1. ` X ) ) |