Step |
Hyp |
Ref |
Expression |
1 |
|
prfcl.p |
|- P = ( F pairF G ) |
2 |
|
prfcl.t |
|- T = ( D Xc. E ) |
3 |
|
prfcl.c |
|- ( ph -> F e. ( C Func D ) ) |
4 |
|
prfcl.d |
|- ( ph -> G e. ( C Func E ) ) |
5 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
6 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
7 |
1 5 6 3 4
|
prfval |
|- ( ph -> P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. ) |
8 |
|
fvex |
|- ( Base ` C ) e. _V |
9 |
8
|
mptex |
|- ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) e. _V |
10 |
8 8
|
mpoex |
|- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) e. _V |
11 |
9 10
|
op1std |
|- ( P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) |
12 |
7 11
|
syl |
|- ( ph -> ( 1st ` P ) = ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) |
13 |
9 10
|
op2ndd |
|- ( P = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. -> ( 2nd ` P ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) ) |
14 |
7 13
|
syl |
|- ( ph -> ( 2nd ` P ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) ) |
15 |
12 14
|
opeq12d |
|- ( ph -> <. ( 1st ` P ) , ( 2nd ` P ) >. = <. ( x e. ( Base ` C ) |-> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) >. ) |
16 |
7 15
|
eqtr4d |
|- ( ph -> P = <. ( 1st ` P ) , ( 2nd ` P ) >. ) |
17 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
18 |
|
eqid |
|- ( Base ` E ) = ( Base ` E ) |
19 |
2 17 18
|
xpcbas |
|- ( ( Base ` D ) X. ( Base ` E ) ) = ( Base ` T ) |
20 |
|
eqid |
|- ( Hom ` T ) = ( Hom ` T ) |
21 |
|
eqid |
|- ( Id ` C ) = ( Id ` C ) |
22 |
|
eqid |
|- ( Id ` T ) = ( Id ` T ) |
23 |
|
eqid |
|- ( comp ` C ) = ( comp ` C ) |
24 |
|
eqid |
|- ( comp ` T ) = ( comp ` T ) |
25 |
|
funcrcl |
|- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) ) |
26 |
3 25
|
syl |
|- ( ph -> ( C e. Cat /\ D e. Cat ) ) |
27 |
26
|
simpld |
|- ( ph -> C e. Cat ) |
28 |
26
|
simprd |
|- ( ph -> D e. Cat ) |
29 |
|
funcrcl |
|- ( G e. ( C Func E ) -> ( C e. Cat /\ E e. Cat ) ) |
30 |
4 29
|
syl |
|- ( ph -> ( C e. Cat /\ E e. Cat ) ) |
31 |
30
|
simprd |
|- ( ph -> E e. Cat ) |
32 |
2 28 31
|
xpccat |
|- ( ph -> T e. Cat ) |
33 |
|
relfunc |
|- Rel ( C Func D ) |
34 |
|
1st2ndbr |
|- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
35 |
33 3 34
|
sylancr |
|- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
36 |
5 17 35
|
funcf1 |
|- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) ) |
37 |
36
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) ) |
38 |
|
relfunc |
|- Rel ( C Func E ) |
39 |
|
1st2ndbr |
|- ( ( Rel ( C Func E ) /\ G e. ( C Func E ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) ) |
40 |
38 4 39
|
sylancr |
|- ( ph -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) ) |
41 |
5 18 40
|
funcf1 |
|- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` E ) ) |
42 |
41
|
ffvelrnda |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) ) |
43 |
37 42
|
opelxpd |
|- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. e. ( ( Base ` D ) X. ( Base ` E ) ) ) |
44 |
12 43
|
fmpt3d |
|- ( ph -> ( 1st ` P ) : ( Base ` C ) --> ( ( Base ` D ) X. ( Base ` E ) ) ) |
45 |
|
eqid |
|- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) |
46 |
|
ovex |
|- ( x ( Hom ` C ) y ) e. _V |
47 |
46
|
mptex |
|- ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) e. _V |
48 |
45 47
|
fnmpoi |
|- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) |
49 |
14
|
fneq1d |
|- ( ph -> ( ( 2nd ` P ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) ) |
50 |
48 49
|
mpbiri |
|- ( ph -> ( 2nd ` P ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) |
51 |
14
|
oveqd |
|- ( ph -> ( x ( 2nd ` P ) y ) = ( x ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) y ) ) |
52 |
45
|
ovmpt4g |
|- ( ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) e. _V ) -> ( x ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) y ) = ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) |
53 |
47 52
|
mp3an3 |
|- ( ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( x ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) y ) = ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) |
54 |
51 53
|
sylan9eq |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` P ) y ) = ( h e. ( x ( Hom ` C ) y ) |-> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. ) ) |
55 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
56 |
35
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
57 |
|
simprl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) ) |
58 |
|
simprr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) ) |
59 |
5 6 55 56 57 58
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) |
60 |
59
|
ffvelrnda |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` F ) y ) ` h ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) |
61 |
|
eqid |
|- ( Hom ` E ) = ( Hom ` E ) |
62 |
40
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) ) |
63 |
5 6 61 62 57 58
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) |
64 |
63
|
ffvelrnda |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` G ) y ) ` h ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) |
65 |
60 64
|
opelxpd |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. e. ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) ) |
66 |
3
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> F e. ( C Func D ) ) |
67 |
4
|
adantr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> G e. ( C Func E ) ) |
68 |
1 5 6 66 67 57
|
prf1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` x ) = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) |
69 |
1 5 6 66 67 58
|
prf1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` P ) ` y ) = <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ) |
70 |
68 69
|
oveq12d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) = ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( Hom ` T ) <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ) ) |
71 |
37
|
adantrr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) ) |
72 |
42
|
adantrr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) ) |
73 |
36
|
ffvelrnda |
|- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) ) |
74 |
73
|
adantrl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) ) |
75 |
41
|
ffvelrnda |
|- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` E ) ) |
76 |
75
|
adantrl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` E ) ) |
77 |
2 17 18 55 61 71 72 74 76 20
|
xpchom2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ( Hom ` T ) <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ) = ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) ) |
78 |
70 77
|
eqtrd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) = ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) ) |
79 |
78
|
adantr |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) = ( ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) X. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) ) |
80 |
65 79
|
eleqtrrd |
|- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ h e. ( x ( Hom ` C ) y ) ) -> <. ( ( x ( 2nd ` F ) y ) ` h ) , ( ( x ( 2nd ` G ) y ) ` h ) >. e. ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) ) |
81 |
54 80
|
fmpt3d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` P ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` P ) ` x ) ( Hom ` T ) ( ( 1st ` P ) ` y ) ) ) |
82 |
|
eqid |
|- ( Id ` D ) = ( Id ` D ) |
83 |
35
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
84 |
|
simpr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) ) |
85 |
5 21 82 83 84
|
funcid |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` F ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) ) |
86 |
|
eqid |
|- ( Id ` E ) = ( Id ` E ) |
87 |
40
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) ) |
88 |
5 21 86 87 84
|
funcid |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) ) |
89 |
85 88
|
opeq12d |
|- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( ( x ( 2nd ` F ) x ) ` ( ( Id ` C ) ` x ) ) , ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) >. = <. ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) , ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) >. ) |
90 |
3
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> F e. ( C Func D ) ) |
91 |
4
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> G e. ( C Func E ) ) |
92 |
27
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> C e. Cat ) |
93 |
5 6 21 92 84
|
catidcl |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` C ) ` x ) e. ( x ( Hom ` C ) x ) ) |
94 |
1 5 6 90 91 84 84 93
|
prf2 |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` P ) x ) ` ( ( Id ` C ) ` x ) ) = <. ( ( x ( 2nd ` F ) x ) ` ( ( Id ` C ) ` x ) ) , ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) >. ) |
95 |
1 5 6 90 91 84
|
prf1 |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` P ) ` x ) = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) |
96 |
95
|
fveq2d |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` T ) ` ( ( 1st ` P ) ` x ) ) = ( ( Id ` T ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) ) |
97 |
28
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat ) |
98 |
31
|
adantr |
|- ( ( ph /\ x e. ( Base ` C ) ) -> E e. Cat ) |
99 |
2 97 98 17 18 82 86 22 37 42
|
xpcid |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` T ) ` <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) = <. ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) , ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) >. ) |
100 |
96 99
|
eqtrd |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` T ) ` ( ( 1st ` P ) ` x ) ) = <. ( ( Id ` D ) ` ( ( 1st ` F ) ` x ) ) , ( ( Id ` E ) ` ( ( 1st ` G ) ` x ) ) >. ) |
101 |
89 94 100
|
3eqtr4d |
|- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( x ( 2nd ` P ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` T ) ` ( ( 1st ` P ) ` x ) ) ) |
102 |
|
eqid |
|- ( comp ` D ) = ( comp ` D ) |
103 |
35
|
3ad2ant1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) ) |
104 |
|
simp21 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> x e. ( Base ` C ) ) |
105 |
|
simp22 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> y e. ( Base ` C ) ) |
106 |
|
simp23 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> z e. ( Base ` C ) ) |
107 |
|
simp3l |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> f e. ( x ( Hom ` C ) y ) ) |
108 |
|
simp3r |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> g e. ( y ( Hom ` C ) z ) ) |
109 |
5 6 23 102 103 104 105 106 107 108
|
funcco |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` F ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) = ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) |
110 |
|
eqid |
|- ( comp ` E ) = ( comp ` E ) |
111 |
4
|
3ad2ant1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> G e. ( C Func E ) ) |
112 |
38 111 39
|
sylancr |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` G ) ( C Func E ) ( 2nd ` G ) ) |
113 |
5 6 23 110 112 104 105 106 107 108
|
funcco |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` G ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) = ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) ) |
114 |
109 113
|
opeq12d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> <. ( ( x ( 2nd ` F ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) , ( ( x ( 2nd ` G ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) >. = <. ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) , ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) >. ) |
115 |
3
|
3ad2ant1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> F e. ( C Func D ) ) |
116 |
27
|
3ad2ant1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> C e. Cat ) |
117 |
5 6 23 116 104 105 106 107 108
|
catcocl |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( g ( <. x , y >. ( comp ` C ) z ) f ) e. ( x ( Hom ` C ) z ) ) |
118 |
1 5 6 115 111 104 106 117
|
prf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` P ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) = <. ( ( x ( 2nd ` F ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) , ( ( x ( 2nd ` G ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) >. ) |
119 |
1 5 6 115 111 104
|
prf1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` P ) ` x ) = <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. ) |
120 |
1 5 6 115 111 105
|
prf1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` P ) ` y ) = <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. ) |
121 |
119 120
|
opeq12d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. = <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. ) |
122 |
1 5 6 115 111 106
|
prf1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` P ) ` z ) = <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. ) |
123 |
121 122
|
oveq12d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. ( comp ` T ) ( ( 1st ` P ) ` z ) ) = ( <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. ( comp ` T ) <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. ) ) |
124 |
1 5 6 115 111 105 106 108
|
prf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( y ( 2nd ` P ) z ) ` g ) = <. ( ( y ( 2nd ` F ) z ) ` g ) , ( ( y ( 2nd ` G ) z ) ` g ) >. ) |
125 |
1 5 6 115 111 104 105 107
|
prf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` P ) y ) ` f ) = <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) |
126 |
123 124 125
|
oveq123d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. ( comp ` T ) ( ( 1st ` P ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) = ( <. ( ( y ( 2nd ` F ) z ) ` g ) , ( ( y ( 2nd ` G ) z ) ` g ) >. ( <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. ( comp ` T ) <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. ) <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) ) |
127 |
36
|
3ad2ant1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) ) |
128 |
127 104
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) ) |
129 |
41
|
3ad2ant1 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` E ) ) |
130 |
129 104
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` E ) ) |
131 |
127 105
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) ) |
132 |
129 105
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` E ) ) |
133 |
127 106
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` F ) ` z ) e. ( Base ` D ) ) |
134 |
129 106
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( 1st ` G ) ` z ) e. ( Base ` E ) ) |
135 |
5 6 55 103 104 105
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) |
136 |
135 107
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) |
137 |
5 6 61 112 104 105
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) |
138 |
137 107
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` G ) y ) ` f ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` E ) ( ( 1st ` G ) ` y ) ) ) |
139 |
5 6 55 103 105 106
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( y ( 2nd ` F ) z ) : ( y ( Hom ` C ) z ) --> ( ( ( 1st ` F ) ` y ) ( Hom ` D ) ( ( 1st ` F ) ` z ) ) ) |
140 |
139 108
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( y ( 2nd ` F ) z ) ` g ) e. ( ( ( 1st ` F ) ` y ) ( Hom ` D ) ( ( 1st ` F ) ` z ) ) ) |
141 |
5 6 61 112 105 106
|
funcf2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( y ( 2nd ` G ) z ) : ( y ( Hom ` C ) z ) --> ( ( ( 1st ` G ) ` y ) ( Hom ` E ) ( ( 1st ` G ) ` z ) ) ) |
142 |
141 108
|
ffvelrnd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( y ( 2nd ` G ) z ) ` g ) e. ( ( ( 1st ` G ) ` y ) ( Hom ` E ) ( ( 1st ` G ) ` z ) ) ) |
143 |
2 17 18 55 61 128 130 131 132 102 110 24 133 134 136 138 140 142
|
xpcco2 |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( <. ( ( y ( 2nd ` F ) z ) ` g ) , ( ( y ( 2nd ` G ) z ) ` g ) >. ( <. <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. , <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. >. ( comp ` T ) <. ( ( 1st ` F ) ` z ) , ( ( 1st ` G ) ` z ) >. ) <. ( ( x ( 2nd ` F ) y ) ` f ) , ( ( x ( 2nd ` G ) y ) ` f ) >. ) = <. ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) , ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) >. ) |
144 |
126 143
|
eqtrd |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. ( comp ` T ) ( ( 1st ` P ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) = <. ( ( ( y ( 2nd ` F ) z ) ` g ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. ( comp ` D ) ( ( 1st ` F ) ` z ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) , ( ( ( y ( 2nd ` G ) z ) ` g ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. ( comp ` E ) ( ( 1st ` G ) ` z ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) >. ) |
145 |
114 118 144
|
3eqtr4d |
|- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ z e. ( Base ` C ) ) /\ ( f e. ( x ( Hom ` C ) y ) /\ g e. ( y ( Hom ` C ) z ) ) ) -> ( ( x ( 2nd ` P ) z ) ` ( g ( <. x , y >. ( comp ` C ) z ) f ) ) = ( ( ( y ( 2nd ` P ) z ) ` g ) ( <. ( ( 1st ` P ) ` x ) , ( ( 1st ` P ) ` y ) >. ( comp ` T ) ( ( 1st ` P ) ` z ) ) ( ( x ( 2nd ` P ) y ) ` f ) ) ) |
146 |
5 19 6 20 21 22 23 24 27 32 44 50 81 101 145
|
isfuncd |
|- ( ph -> ( 1st ` P ) ( C Func T ) ( 2nd ` P ) ) |
147 |
|
df-br |
|- ( ( 1st ` P ) ( C Func T ) ( 2nd ` P ) <-> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( C Func T ) ) |
148 |
146 147
|
sylib |
|- ( ph -> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( C Func T ) ) |
149 |
16 148
|
eqeltrd |
|- ( ph -> P e. ( C Func T ) ) |