Step |
Hyp |
Ref |
Expression |
1 |
|
catcoppccl.c |
|- C = ( CatCat ` U ) |
2 |
|
catcoppccl.b |
|- B = ( Base ` C ) |
3 |
|
catcoppccl.o |
|- O = ( oppCat ` X ) |
4 |
|
catcoppccl.1 |
|- ( ph -> U e. WUni ) |
5 |
|
catcoppccl.2 |
|- ( ph -> _om e. U ) |
6 |
|
catcoppccl.3 |
|- ( ph -> X e. B ) |
7 |
|
eqid |
|- ( Base ` X ) = ( Base ` X ) |
8 |
|
eqid |
|- ( Hom ` X ) = ( Hom ` X ) |
9 |
|
eqid |
|- ( comp ` X ) = ( comp ` X ) |
10 |
7 8 9 3
|
oppcval |
|- ( X e. B -> O = ( ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) sSet <. ( comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) >. ) ) |
11 |
6 10
|
syl |
|- ( ph -> O = ( ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) sSet <. ( comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) >. ) ) |
12 |
1 2 4
|
catcbas |
|- ( ph -> B = ( U i^i Cat ) ) |
13 |
6 12
|
eleqtrd |
|- ( ph -> X e. ( U i^i Cat ) ) |
14 |
13
|
elin1d |
|- ( ph -> X e. U ) |
15 |
|
df-hom |
|- Hom = Slot ; 1 4 |
16 |
4 5
|
wunndx |
|- ( ph -> ndx e. U ) |
17 |
15 4 16
|
wunstr |
|- ( ph -> ( Hom ` ndx ) e. U ) |
18 |
15 4 14
|
wunstr |
|- ( ph -> ( Hom ` X ) e. U ) |
19 |
4 18
|
wuntpos |
|- ( ph -> tpos ( Hom ` X ) e. U ) |
20 |
4 17 19
|
wunop |
|- ( ph -> <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. e. U ) |
21 |
4 14 20
|
wunsets |
|- ( ph -> ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) e. U ) |
22 |
|
df-cco |
|- comp = Slot ; 1 5 |
23 |
22 4 16
|
wunstr |
|- ( ph -> ( comp ` ndx ) e. U ) |
24 |
|
df-base |
|- Base = Slot 1 |
25 |
24 4 14
|
wunstr |
|- ( ph -> ( Base ` X ) e. U ) |
26 |
4 25 25
|
wunxp |
|- ( ph -> ( ( Base ` X ) X. ( Base ` X ) ) e. U ) |
27 |
4 26 25
|
wunxp |
|- ( ph -> ( ( ( Base ` X ) X. ( Base ` X ) ) X. ( Base ` X ) ) e. U ) |
28 |
22 4 14
|
wunstr |
|- ( ph -> ( comp ` X ) e. U ) |
29 |
4 28
|
wunrn |
|- ( ph -> ran ( comp ` X ) e. U ) |
30 |
4 29
|
wununi |
|- ( ph -> U. ran ( comp ` X ) e. U ) |
31 |
4 30
|
wundm |
|- ( ph -> dom U. ran ( comp ` X ) e. U ) |
32 |
4 31
|
wuncnv |
|- ( ph -> `' dom U. ran ( comp ` X ) e. U ) |
33 |
4
|
wun0 |
|- ( ph -> (/) e. U ) |
34 |
4 33
|
wunsn |
|- ( ph -> { (/) } e. U ) |
35 |
4 32 34
|
wunun |
|- ( ph -> ( `' dom U. ran ( comp ` X ) u. { (/) } ) e. U ) |
36 |
4 30
|
wunrn |
|- ( ph -> ran U. ran ( comp ` X ) e. U ) |
37 |
4 35 36
|
wunxp |
|- ( ph -> ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) e. U ) |
38 |
4 37
|
wunpw |
|- ( ph -> ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) e. U ) |
39 |
|
tposssxp |
|- tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) u. { (/) } ) X. ran ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) |
40 |
|
ovssunirn |
|- ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ U. ran ( comp ` X ) |
41 |
|
dmss |
|- ( ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ U. ran ( comp ` X ) -> dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ dom U. ran ( comp ` X ) ) |
42 |
40 41
|
ax-mp |
|- dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ dom U. ran ( comp ` X ) |
43 |
|
cnvss |
|- ( dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ dom U. ran ( comp ` X ) -> `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ `' dom U. ran ( comp ` X ) ) |
44 |
|
unss1 |
|- ( `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ `' dom U. ran ( comp ` X ) -> ( `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) u. { (/) } ) C_ ( `' dom U. ran ( comp ` X ) u. { (/) } ) ) |
45 |
42 43 44
|
mp2b |
|- ( `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) u. { (/) } ) C_ ( `' dom U. ran ( comp ` X ) u. { (/) } ) |
46 |
40
|
rnssi |
|- ran ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ ran U. ran ( comp ` X ) |
47 |
|
xpss12 |
|- ( ( ( `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) u. { (/) } ) C_ ( `' dom U. ran ( comp ` X ) u. { (/) } ) /\ ran ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ ran U. ran ( comp ` X ) ) -> ( ( `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) u. { (/) } ) X. ran ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) C_ ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) |
48 |
45 46 47
|
mp2an |
|- ( ( `' dom ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) u. { (/) } ) X. ran ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) C_ ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) |
49 |
39 48
|
sstri |
|- tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) |
50 |
|
elpw2g |
|- ( ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) e. U -> ( tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) <-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) ) |
51 |
37 50
|
syl |
|- ( ph -> ( tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) <-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) C_ ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) ) |
52 |
49 51
|
mpbiri |
|- ( ph -> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) |
53 |
52
|
ralrimivw |
|- ( ph -> A. y e. ( Base ` X ) tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) |
54 |
53
|
ralrimivw |
|- ( ph -> A. x e. ( ( Base ` X ) X. ( Base ` X ) ) A. y e. ( Base ` X ) tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) |
55 |
|
eqid |
|- ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) = ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) |
56 |
55
|
fmpo |
|- ( A. x e. ( ( Base ` X ) X. ( Base ` X ) ) A. y e. ( Base ` X ) tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) e. ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) <-> ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) : ( ( ( Base ` X ) X. ( Base ` X ) ) X. ( Base ` X ) ) --> ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) |
57 |
54 56
|
sylib |
|- ( ph -> ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) : ( ( ( Base ` X ) X. ( Base ` X ) ) X. ( Base ` X ) ) --> ~P ( ( `' dom U. ran ( comp ` X ) u. { (/) } ) X. ran U. ran ( comp ` X ) ) ) |
58 |
4 27 38 57
|
wunf |
|- ( ph -> ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) e. U ) |
59 |
4 23 58
|
wunop |
|- ( ph -> <. ( comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) >. e. U ) |
60 |
4 21 59
|
wunsets |
|- ( ph -> ( ( X sSet <. ( Hom ` ndx ) , tpos ( Hom ` X ) >. ) sSet <. ( comp ` ndx ) , ( x e. ( ( Base ` X ) X. ( Base ` X ) ) , y e. ( Base ` X ) |-> tpos ( <. y , ( 2nd ` x ) >. ( comp ` X ) ( 1st ` x ) ) ) >. ) e. U ) |
61 |
11 60
|
eqeltrd |
|- ( ph -> O e. U ) |
62 |
13
|
elin2d |
|- ( ph -> X e. Cat ) |
63 |
3
|
oppccat |
|- ( X e. Cat -> O e. Cat ) |
64 |
62 63
|
syl |
|- ( ph -> O e. Cat ) |
65 |
61 64
|
elind |
|- ( ph -> O e. ( U i^i Cat ) ) |
66 |
65 12
|
eleqtrrd |
|- ( ph -> O e. B ) |