Step |
Hyp |
Ref |
Expression |
1 |
|
metuust |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( metUnif ` D ) e. ( UnifOn ` X ) ) |
2 |
|
utopval |
|- ( ( metUnif ` D ) e. ( UnifOn ` X ) -> ( unifTop ` ( metUnif ` D ) ) = { a e. ~P X | A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a } ) |
3 |
1 2
|
syl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( unifTop ` ( metUnif ` D ) ) = { a e. ~P X | A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a } ) |
4 |
3
|
eleq2d |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( a e. ( unifTop ` ( metUnif ` D ) ) <-> a e. { a e. ~P X | A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a } ) ) |
5 |
|
rabid |
|- ( a e. { a e. ~P X | A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a } <-> ( a e. ~P X /\ A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) |
6 |
4 5
|
bitrdi |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( a e. ( unifTop ` ( metUnif ` D ) ) <-> ( a e. ~P X /\ A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) ) |
7 |
6
|
biimpa |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> ( a e. ~P X /\ A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) |
8 |
7
|
simpld |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> a e. ~P X ) |
9 |
8
|
elpwid |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> a C_ X ) |
10 |
|
unirnblps |
|- ( D e. ( PsMet ` X ) -> U. ran ( ball ` D ) = X ) |
11 |
10
|
ad2antlr |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> U. ran ( ball ` D ) = X ) |
12 |
9 11
|
sseqtrrd |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> a C_ U. ran ( ball ` D ) ) |
13 |
|
simpr |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> ( v " { x } ) C_ a ) |
14 |
|
simp-5r |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> D e. ( PsMet ` X ) ) |
15 |
|
simplr |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> v e. ( metUnif ` D ) ) |
16 |
9
|
ad3antrrr |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> a C_ X ) |
17 |
|
simpllr |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> x e. a ) |
18 |
16 17
|
sseldd |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> x e. X ) |
19 |
|
metustbl |
|- ( ( D e. ( PsMet ` X ) /\ v e. ( metUnif ` D ) /\ x e. X ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ ( v " { x } ) ) ) |
20 |
14 15 18 19
|
syl3anc |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ ( v " { x } ) ) ) |
21 |
|
sstr |
|- ( ( b C_ ( v " { x } ) /\ ( v " { x } ) C_ a ) -> b C_ a ) |
22 |
21
|
expcom |
|- ( ( v " { x } ) C_ a -> ( b C_ ( v " { x } ) -> b C_ a ) ) |
23 |
22
|
anim2d |
|- ( ( v " { x } ) C_ a -> ( ( x e. b /\ b C_ ( v " { x } ) ) -> ( x e. b /\ b C_ a ) ) ) |
24 |
23
|
reximdv |
|- ( ( v " { x } ) C_ a -> ( E. b e. ran ( ball ` D ) ( x e. b /\ b C_ ( v " { x } ) ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) |
25 |
13 20 24
|
sylc |
|- ( ( ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) /\ v e. ( metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) |
26 |
7
|
simprd |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) |
27 |
26
|
r19.21bi |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) -> E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) |
28 |
25 27
|
r19.29a |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) /\ x e. a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) |
29 |
28
|
ralrimiva |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) |
30 |
12 29
|
jca |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) |
31 |
|
fvex |
|- ( ball ` D ) e. _V |
32 |
31
|
rnex |
|- ran ( ball ` D ) e. _V |
33 |
|
eltg2 |
|- ( ran ( ball ` D ) e. _V -> ( a e. ( topGen ` ran ( ball ` D ) ) <-> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) ) |
34 |
32 33
|
mp1i |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> ( a e. ( topGen ` ran ( ball ` D ) ) <-> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) ) |
35 |
30 34
|
mpbird |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( unifTop ` ( metUnif ` D ) ) ) -> a e. ( topGen ` ran ( ball ` D ) ) ) |
36 |
32 33
|
mp1i |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( a e. ( topGen ` ran ( ball ` D ) ) <-> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) ) |
37 |
36
|
biimpa |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) |
38 |
37
|
simpld |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ U. ran ( ball ` D ) ) |
39 |
10
|
ad2antlr |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> U. ran ( ball ` D ) = X ) |
40 |
38 39
|
sseqtrd |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ X ) |
41 |
|
elpwg |
|- ( a e. ( topGen ` ran ( ball ` D ) ) -> ( a e. ~P X <-> a C_ X ) ) |
42 |
41
|
adantl |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a e. ~P X <-> a C_ X ) ) |
43 |
40 42
|
mpbird |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. ~P X ) |
44 |
|
simpllr |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> D e. ( PsMet ` X ) ) |
45 |
40
|
sselda |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> x e. X ) |
46 |
37
|
simprd |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) |
47 |
46
|
r19.21bi |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) |
48 |
|
blssexps |
|- ( ( D e. ( PsMet ` X ) /\ x e. X ) -> ( E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) <-> E. d e. RR+ ( x ( ball ` D ) d ) C_ a ) ) |
49 |
44 45 48
|
syl2anc |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> ( E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) <-> E. d e. RR+ ( x ( ball ` D ) d ) C_ a ) ) |
50 |
47 49
|
mpbid |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. d e. RR+ ( x ( ball ` D ) d ) C_ a ) |
51 |
|
blval2 |
|- ( ( D e. ( PsMet ` X ) /\ x e. X /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) ) |
52 |
51
|
3expa |
|- ( ( ( D e. ( PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) ) |
53 |
52
|
sseq1d |
|- ( ( ( D e. ( PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( ( x ( ball ` D ) d ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) ) |
54 |
53
|
rexbidva |
|- ( ( D e. ( PsMet ` X ) /\ x e. X ) -> ( E. d e. RR+ ( x ( ball ` D ) d ) C_ a <-> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) ) |
55 |
54
|
biimpa |
|- ( ( ( D e. ( PsMet ` X ) /\ x e. X ) /\ E. d e. RR+ ( x ( ball ` D ) d ) C_ a ) -> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) |
56 |
44 45 50 55
|
syl21anc |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) |
57 |
|
cnvexg |
|- ( D e. ( PsMet ` X ) -> `' D e. _V ) |
58 |
|
imaexg |
|- ( `' D e. _V -> ( `' D " ( 0 [,) d ) ) e. _V ) |
59 |
57 58
|
syl |
|- ( D e. ( PsMet ` X ) -> ( `' D " ( 0 [,) d ) ) e. _V ) |
60 |
59
|
ralrimivw |
|- ( D e. ( PsMet ` X ) -> A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V ) |
61 |
|
eqid |
|- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) |
62 |
|
imaeq1 |
|- ( v = ( `' D " ( 0 [,) d ) ) -> ( v " { x } ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) ) |
63 |
62
|
sseq1d |
|- ( v = ( `' D " ( 0 [,) d ) ) -> ( ( v " { x } ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) ) |
64 |
61 63
|
rexrnmptw |
|- ( A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a <-> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) ) |
65 |
44 60 64
|
3syl |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a <-> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) ) |
66 |
56 65
|
mpbird |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a ) |
67 |
|
oveq2 |
|- ( d = e -> ( 0 [,) d ) = ( 0 [,) e ) ) |
68 |
67
|
imaeq2d |
|- ( d = e -> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) e ) ) ) |
69 |
68
|
cbvmptv |
|- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) ) |
70 |
69
|
rneqi |
|- ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ran ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) ) |
71 |
70
|
metustfbas |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) ) |
72 |
|
ssfg |
|- ( ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) ) |
73 |
71 72
|
syl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) ) |
74 |
|
metuval |
|- ( D e. ( PsMet ` X ) -> ( metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) ) |
75 |
74
|
adantl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) ) |
76 |
73 75
|
sseqtrrd |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ ( metUnif ` D ) ) |
77 |
|
ssrexv |
|- ( ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ ( metUnif ` D ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a -> E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) |
78 |
76 77
|
syl |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a -> E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) |
79 |
78
|
ad2antrr |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a -> E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) |
80 |
66 79
|
mpd |
|- ( ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) |
81 |
80
|
ralrimiva |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) |
82 |
43 81
|
jca |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a e. ~P X /\ A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) |
83 |
6
|
biimpar |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ ( a e. ~P X /\ A. x e. a E. v e. ( metUnif ` D ) ( v " { x } ) C_ a ) ) -> a e. ( unifTop ` ( metUnif ` D ) ) ) |
84 |
82 83
|
syldan |
|- ( ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. ( unifTop ` ( metUnif ` D ) ) ) |
85 |
35 84
|
impbida |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( a e. ( unifTop ` ( metUnif ` D ) ) <-> a e. ( topGen ` ran ( ball ` D ) ) ) ) |
86 |
85
|
eqrdv |
|- ( ( X =/= (/) /\ D e. ( PsMet ` X ) ) -> ( unifTop ` ( metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) ) |