Step |
Hyp |
Ref |
Expression |
1 |
|
resscatc.c |
|- C = ( CatCat ` U ) |
2 |
|
resscatc.d |
|- D = ( CatCat ` V ) |
3 |
|
resscatc.1 |
|- ( ph -> U e. W ) |
4 |
|
resscatc.2 |
|- ( ph -> V C_ U ) |
5 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
6 |
3 4
|
ssexd |
|- ( ph -> V e. _V ) |
7 |
6
|
adantr |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> V e. _V ) |
8 |
|
eqid |
|- ( Hom ` D ) = ( Hom ` D ) |
9 |
|
simprl |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> x e. ( V i^i Cat ) ) |
10 |
2 5 6
|
catcbas |
|- ( ph -> ( Base ` D ) = ( V i^i Cat ) ) |
11 |
10
|
adantr |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( Base ` D ) = ( V i^i Cat ) ) |
12 |
9 11
|
eleqtrrd |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> x e. ( Base ` D ) ) |
13 |
|
simprr |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> y e. ( V i^i Cat ) ) |
14 |
13 11
|
eleqtrrd |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> y e. ( Base ` D ) ) |
15 |
2 5 7 8 12 14
|
catchom |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` D ) y ) = ( x Func y ) ) |
16 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
17 |
3
|
adantr |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> U e. W ) |
18 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
19 |
|
inass |
|- ( ( V i^i U ) i^i Cat ) = ( V i^i ( U i^i Cat ) ) |
20 |
1 16 3
|
catcbas |
|- ( ph -> ( Base ` C ) = ( U i^i Cat ) ) |
21 |
20
|
ineq2d |
|- ( ph -> ( V i^i ( Base ` C ) ) = ( V i^i ( U i^i Cat ) ) ) |
22 |
19 21
|
eqtr4id |
|- ( ph -> ( ( V i^i U ) i^i Cat ) = ( V i^i ( Base ` C ) ) ) |
23 |
|
df-ss |
|- ( V C_ U <-> ( V i^i U ) = V ) |
24 |
4 23
|
sylib |
|- ( ph -> ( V i^i U ) = V ) |
25 |
24
|
ineq1d |
|- ( ph -> ( ( V i^i U ) i^i Cat ) = ( V i^i Cat ) ) |
26 |
|
eqid |
|- ( C |`s V ) = ( C |`s V ) |
27 |
26 16
|
ressbas |
|- ( V e. _V -> ( V i^i ( Base ` C ) ) = ( Base ` ( C |`s V ) ) ) |
28 |
6 27
|
syl |
|- ( ph -> ( V i^i ( Base ` C ) ) = ( Base ` ( C |`s V ) ) ) |
29 |
22 25 28
|
3eqtr3d |
|- ( ph -> ( V i^i Cat ) = ( Base ` ( C |`s V ) ) ) |
30 |
26 16
|
ressbasss |
|- ( Base ` ( C |`s V ) ) C_ ( Base ` C ) |
31 |
29 30
|
eqsstrdi |
|- ( ph -> ( V i^i Cat ) C_ ( Base ` C ) ) |
32 |
31
|
adantr |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( V i^i Cat ) C_ ( Base ` C ) ) |
33 |
32 9
|
sseldd |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> x e. ( Base ` C ) ) |
34 |
32 13
|
sseldd |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> y e. ( Base ` C ) ) |
35 |
1 16 17 18 33 34
|
catchom |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` C ) y ) = ( x Func y ) ) |
36 |
26 18
|
resshom |
|- ( V e. _V -> ( Hom ` C ) = ( Hom ` ( C |`s V ) ) ) |
37 |
6 36
|
syl |
|- ( ph -> ( Hom ` C ) = ( Hom ` ( C |`s V ) ) ) |
38 |
37
|
oveqdr |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` C ) y ) = ( x ( Hom ` ( C |`s V ) ) y ) ) |
39 |
15 35 38
|
3eqtr2rd |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) ) ) -> ( x ( Hom ` ( C |`s V ) ) y ) = ( x ( Hom ` D ) y ) ) |
40 |
39
|
ralrimivva |
|- ( ph -> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) ( x ( Hom ` ( C |`s V ) ) y ) = ( x ( Hom ` D ) y ) ) |
41 |
|
eqid |
|- ( Hom ` ( C |`s V ) ) = ( Hom ` ( C |`s V ) ) |
42 |
10
|
eqcomd |
|- ( ph -> ( V i^i Cat ) = ( Base ` D ) ) |
43 |
41 8 29 42
|
homfeq |
|- ( ph -> ( ( Homf ` ( C |`s V ) ) = ( Homf ` D ) <-> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) ( x ( Hom ` ( C |`s V ) ) y ) = ( x ( Hom ` D ) y ) ) ) |
44 |
40 43
|
mpbird |
|- ( ph -> ( Homf ` ( C |`s V ) ) = ( Homf ` D ) ) |
45 |
6
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> V e. _V ) |
46 |
|
eqid |
|- ( comp ` D ) = ( comp ` D ) |
47 |
|
simplr1 |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> x e. ( V i^i Cat ) ) |
48 |
10
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( Base ` D ) = ( V i^i Cat ) ) |
49 |
47 48
|
eleqtrrd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> x e. ( Base ` D ) ) |
50 |
|
simplr2 |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> y e. ( V i^i Cat ) ) |
51 |
50 48
|
eleqtrrd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> y e. ( Base ` D ) ) |
52 |
|
simplr3 |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> z e. ( V i^i Cat ) ) |
53 |
52 48
|
eleqtrrd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> z e. ( Base ` D ) ) |
54 |
|
simprl |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> f e. ( x ( Hom ` D ) y ) ) |
55 |
2 5 45 8 49 51
|
catchom |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( x ( Hom ` D ) y ) = ( x Func y ) ) |
56 |
54 55
|
eleqtrd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> f e. ( x Func y ) ) |
57 |
|
simprr |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> g e. ( y ( Hom ` D ) z ) ) |
58 |
2 5 45 8 51 53
|
catchom |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( y ( Hom ` D ) z ) = ( y Func z ) ) |
59 |
57 58
|
eleqtrd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> g e. ( y Func z ) ) |
60 |
2 5 45 46 49 51 53 56 59
|
catcco |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. ( comp ` D ) z ) f ) = ( g o.func f ) ) |
61 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> U e. W ) |
62 |
|
eqid |
|- ( comp ` C ) = ( comp ` C ) |
63 |
31
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( V i^i Cat ) C_ ( Base ` C ) ) |
64 |
63 47
|
sseldd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> x e. ( Base ` C ) ) |
65 |
63 50
|
sseldd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> y e. ( Base ` C ) ) |
66 |
63 52
|
sseldd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> z e. ( Base ` C ) ) |
67 |
1 16 61 62 64 65 66 56 59
|
catcco |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. ( comp ` C ) z ) f ) = ( g o.func f ) ) |
68 |
26 62
|
ressco |
|- ( V e. _V -> ( comp ` C ) = ( comp ` ( C |`s V ) ) ) |
69 |
6 68
|
syl |
|- ( ph -> ( comp ` C ) = ( comp ` ( C |`s V ) ) ) |
70 |
69
|
ad2antrr |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( comp ` C ) = ( comp ` ( C |`s V ) ) ) |
71 |
70
|
oveqd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( <. x , y >. ( comp ` C ) z ) = ( <. x , y >. ( comp ` ( C |`s V ) ) z ) ) |
72 |
71
|
oveqd |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. ( comp ` C ) z ) f ) = ( g ( <. x , y >. ( comp ` ( C |`s V ) ) z ) f ) ) |
73 |
60 67 72
|
3eqtr2d |
|- ( ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) /\ ( f e. ( x ( Hom ` D ) y ) /\ g e. ( y ( Hom ` D ) z ) ) ) -> ( g ( <. x , y >. ( comp ` D ) z ) f ) = ( g ( <. x , y >. ( comp ` ( C |`s V ) ) z ) f ) ) |
74 |
73
|
ralrimivva |
|- ( ( ph /\ ( x e. ( V i^i Cat ) /\ y e. ( V i^i Cat ) /\ z e. ( V i^i Cat ) ) ) -> A. f e. ( x ( Hom ` D ) y ) A. g e. ( y ( Hom ` D ) z ) ( g ( <. x , y >. ( comp ` D ) z ) f ) = ( g ( <. x , y >. ( comp ` ( C |`s V ) ) z ) f ) ) |
75 |
74
|
ralrimivvva |
|- ( ph -> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) A. z e. ( V i^i Cat ) A. f e. ( x ( Hom ` D ) y ) A. g e. ( y ( Hom ` D ) z ) ( g ( <. x , y >. ( comp ` D ) z ) f ) = ( g ( <. x , y >. ( comp ` ( C |`s V ) ) z ) f ) ) |
76 |
|
eqid |
|- ( comp ` ( C |`s V ) ) = ( comp ` ( C |`s V ) ) |
77 |
44
|
eqcomd |
|- ( ph -> ( Homf ` D ) = ( Homf ` ( C |`s V ) ) ) |
78 |
46 76 8 42 29 77
|
comfeq |
|- ( ph -> ( ( comf ` D ) = ( comf ` ( C |`s V ) ) <-> A. x e. ( V i^i Cat ) A. y e. ( V i^i Cat ) A. z e. ( V i^i Cat ) A. f e. ( x ( Hom ` D ) y ) A. g e. ( y ( Hom ` D ) z ) ( g ( <. x , y >. ( comp ` D ) z ) f ) = ( g ( <. x , y >. ( comp ` ( C |`s V ) ) z ) f ) ) ) |
79 |
75 78
|
mpbird |
|- ( ph -> ( comf ` D ) = ( comf ` ( C |`s V ) ) ) |
80 |
79
|
eqcomd |
|- ( ph -> ( comf ` ( C |`s V ) ) = ( comf ` D ) ) |
81 |
44 80
|
jca |
|- ( ph -> ( ( Homf ` ( C |`s V ) ) = ( Homf ` D ) /\ ( comf ` ( C |`s V ) ) = ( comf ` D ) ) ) |