Step |
Hyp |
Ref |
Expression |
1 |
|
fuccocl.q |
|- Q = ( C FuncCat D ) |
2 |
|
fuccocl.n |
|- N = ( C Nat D ) |
3 |
|
fuccocl.x |
|- .xb = ( comp ` Q ) |
4 |
|
fuccocl.r |
|- ( ph -> R e. ( F N G ) ) |
5 |
|
fuccocl.s |
|- ( ph -> S e. ( G N H ) ) |
6 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
7 |
|
eqid |
|- ( comp ` D ) = ( comp ` D ) |
8 |
1 2 6 7 3 4 5
|
fucco |
|- ( ph -> ( S ( <. F , G >. .xb H ) R ) = ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) ) |
9 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
10 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
11 |
2
|
natrcl |
|- ( R e. ( F N G ) -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) ) |
12 |
4 11
|
syl |
|- ( ph -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) ) |
13 |
12
|
simpld |
|- ( ph -> F e. ( C Func D ) ) |
14 |
|
funcrcl |
|- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) ) |
15 |
13 14
|
syl |
|- ( ph -> ( C e. Cat /\ D e. Cat ) ) |
16 |
15
|
simprd |
|- ( ph -> D e. Cat ) |
17 |
16
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat ) |
18 |
|
relfunc |
|- Rel ( C Func D ) |
19 |
|
1st2ndbr |
|- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
20 |
18 13 19
|
sylancr |
|- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
21 |
6 9 20
|
funcf1 |
|- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) ) |
22 |
21
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) ) |
23 |
2
|
natrcl |
|- ( S e. ( G N H ) -> ( G e. ( C Func D ) /\ H e. ( C Func D ) ) ) |
24 |
5 23
|
syl |
|- ( ph -> ( G e. ( C Func D ) /\ H e. ( C Func D ) ) ) |
25 |
24
|
simpld |
|- ( ph -> G e. ( C Func D ) ) |
26 |
|
1st2ndbr |
|- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) ) |
27 |
18 25 26
|
sylancr |
|- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) ) |
28 |
6 9 27
|
funcf1 |
|- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) ) |
29 |
28
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) ) |
30 |
24
|
simprd |
|- ( ph -> H e. ( C Func D ) ) |
31 |
|
1st2ndbr |
|- ( ( Rel ( C Func D ) /\ H e. ( C Func D ) ) -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) ) |
32 |
18 30 31
|
sylancr |
|- ( ph -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) ) |
33 |
6 9 32
|
funcf1 |
|- ( ph -> ( 1st ` H ) : ( Base ` C ) --> ( Base ` D ) ) |
34 |
33
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` H ) ` x ) e. ( Base ` D ) ) |
35 |
2 4
|
nat1st2nd |
|- ( ph -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) ) |
36 |
35
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) ) |
37 |
|
simpr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) ) |
38 |
2 36 6 10 37
|
natcl |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( R ` x ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` x ) ) ) |
39 |
2 5
|
nat1st2nd |
|- ( ph -> S e. ( <. ( 1st ` G ) , ( 2nd ` G ) >. N <. ( 1st ` H ) , ( 2nd ` H ) >. ) ) |
40 |
39
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> S e. ( <. ( 1st ` G ) , ( 2nd ` G ) >. N <. ( 1st ` H ) , ( 2nd ` H ) >. ) ) |
41 |
2 40 6 10 37
|
natcl |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( S ` x ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) |
42 |
9 10 7 17 22 29 34 38 41
|
catcocl |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) |
43 |
42
|
ralrimiva |
|- ( ph -> A. x e. ( Base ` C ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) |
44 |
|
fvex |
|- ( Base ` C ) e. _V |
45 |
|
mptelixpg |
|- ( ( Base ` C ) e. _V -> ( ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) <-> A. x e. ( Base ` C ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) ) |
46 |
44 45
|
ax-mp |
|- ( ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) <-> A. x e. ( Base ` C ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) |
47 |
43 46
|
sylibr |
|- ( ph -> ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) |
48 |
8 47
|
eqeltrd |
|- ( ph -> ( S ( <. F , G >. .xb H ) R ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) |
49 |
16
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> D e. Cat ) |
50 |
21
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) ) |
51 |
|
simpr1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> x e. ( Base ` C ) ) |
52 |
50 51
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) ) |
53 |
|
simpr2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> y e. ( Base ` C ) ) |
54 |
50 53
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) ) |
55 |
28
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) ) |
56 |
55 53
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` D ) ) |
57 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
58 |
20
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
59 |
6 57 10 58 51 53
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) |
60 |
|
simpr3 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> f e. ( x ( Hom ` C ) y ) ) |
61 |
59 60
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) |
62 |
35
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) ) |
63 |
2 62 6 10 53
|
natcl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( R ` y ) e. ( ( ( 1st ` F ) ` y ) ( Hom ` D ) ( ( 1st ` G ) ` y ) ) ) |
64 |
33
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` H ) : ( Base ` C ) --> ( Base ` D ) ) |
65 |
64 53
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` H ) ` y ) e. ( Base ` D ) ) |
66 |
39
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> S e. ( <. ( 1st ` G ) , ( 2nd ` G ) >. N <. ( 1st ` H ) , ( 2nd ` H ) >. ) ) |
67 |
2 66 6 10 53
|
natcl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( S ` y ) e. ( ( ( 1st ` G ) ` y ) ( Hom ` D ) ( ( 1st ` H ) ` y ) ) ) |
68 |
9 10 7 49 52 54 56 61 63 65 67
|
catass |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) ) |
69 |
2 62 6 57 7 51 53 60
|
nati |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` G ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` G ) ` y ) ) ( R ` x ) ) ) |
70 |
69
|
oveq2d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( ( x ( 2nd ` G ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` G ) ` y ) ) ( R ` x ) ) ) ) |
71 |
55 51
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) ) |
72 |
2 62 6 10 51
|
natcl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( R ` x ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` x ) ) ) |
73 |
27
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) ) |
74 |
6 57 10 73 51 53
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` y ) ) ) |
75 |
74 60
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( x ( 2nd ` G ) y ) ` f ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` y ) ) ) |
76 |
9 10 7 49 52 71 56 72 75 65 67
|
catass |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( ( x ( 2nd ` G ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` G ) ` y ) ) ( R ` x ) ) ) ) |
77 |
2 66 6 57 7 51 53 60
|
nati |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ` y ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) ) |
78 |
77
|
oveq1d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) = ( ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) ) |
79 |
70 76 78
|
3eqtr2d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) = ( ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) ) |
80 |
64 51
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` H ) ` x ) e. ( Base ` D ) ) |
81 |
2 66 6 10 51
|
natcl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( S ` x ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) |
82 |
32
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) ) |
83 |
6 57 10 82 51 53
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( x ( 2nd ` H ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` H ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` y ) ) ) |
84 |
83 60
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( x ( 2nd ` H ) y ) ` f ) e. ( ( ( 1st ` H ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` y ) ) ) |
85 |
9 10 7 49 52 71 80 72 81 65 84
|
catass |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) ) |
86 |
68 79 85
|
3eqtrd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) ) |
87 |
4
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> R e. ( F N G ) ) |
88 |
5
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> S e. ( G N H ) ) |
89 |
1 2 6 7 3 87 88 53
|
fuccoval |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ( <. F , G >. .xb H ) R ) ` y ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) ) |
90 |
89
|
oveq1d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) |
91 |
1 2 6 7 3 87 88 51
|
fuccoval |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ( <. F , G >. .xb H ) R ) ` x ) = ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) |
92 |
91
|
oveq2d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) ) |
93 |
86 90 92
|
3eqtr4d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) ) |
94 |
93
|
ralrimivvva |
|- ( ph -> A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) ) |
95 |
2 6 57 10 7 13 30
|
isnat2 |
|- ( ph -> ( ( S ( <. F , G >. .xb H ) R ) e. ( F N H ) <-> ( ( S ( <. F , G >. .xb H ) R ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) /\ A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. ( comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) ) ) ) |
96 |
48 94 95
|
mpbir2and |
|- ( ph -> ( S ( <. F , G >. .xb H ) R ) e. ( F N H ) ) |