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Theorem psmetutop 20814
Description: The topology induced by a uniform structure generated by a metric  D is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
psmetutop  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (unifTop `  (metUnif `  D ) )  =  ( topGen `  ran  ( ball `  D ) ) )

Proof of Theorem psmetutop
Dummy variables  a 
b  d  e  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metuust 20803 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
2 utopval 20463 . . . . . . . . . . . 12  |-  ( (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 }
)
31, 2syl 16 . . . . . . . . . . 11  |-  ( ( 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 } )
43eleq2d 2530 . . . . . . . . . 10  |-  ( ( 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 3031 . . . . . . . . . 10  |-  ( 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 ) )
64, 5syl6bb 261 . . . . . . . . 9  |-  ( ( 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 ) ) )
76biimpa 484 . . . . . . . 8  |-  ( ( ( 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 ) )
87simpld 459 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ~P X
)
98elpwid 4013 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  X )
10 unirnblps 20650 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
1110ad2antlr 726 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  U. ran  ( ball `  D
)  =  X )
129, 11sseqtr4d 3534 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  U. ran  ( ball `  D ) )
13 simpr 461 . . . . . . . 8  |-  ( ( ( ( ( ( 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 768 . . . . . . . . 9  |-  ( ( ( ( ( ( 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 754 . . . . . . . . 9  |-  ( ( ( ( ( ( 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 )
)
169ad3antrrr 729 . . . . . . . . . 10  |-  ( ( ( ( ( ( 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 758 . . . . . . . . . 10  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnif `  D ) )  /\  ( v " {
x } )  C_  a )  ->  x  e.  a )
1816, 17sseldd 3498 . . . . . . . . 9  |-  ( ( ( ( ( ( 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 20812 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  v  e.  (metUnif `  D )  /\  x  e.  X
)  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  ( v " {
x } ) ) )
2014, 15, 18, 19syl3anc 1223 . . . . . . . 8  |-  ( ( ( ( ( ( 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 3505 . . . . . . . . . . 11  |-  ( ( b  C_  ( v " { x } )  /\  ( v " { x } ) 
C_  a )  -> 
b  C_  a )
2221expcom 435 . . . . . . . . . 10  |-  ( ( v " { x } )  C_  a  ->  ( b  C_  (
v " { x } )  ->  b  C_  a ) )
2322anim2d 565 . . . . . . . . 9  |-  ( ( v " { x } )  C_  a  ->  ( ( x  e.  b  /\  b  C_  ( v " {
x } ) )  ->  ( x  e.  b  /\  b  C_  a ) ) )
2423reximdv 2930 . . . . . . . 8  |-  ( ( 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
) ) )
2513, 20, 24sylc 60 . . . . . . 7  |-  ( ( ( ( ( ( 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 ) )
267simprd 463 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  A. x  e.  a  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a )
2726r19.21bi 2826 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  (unifTop `  (metUnif `  D ) ) )  /\  x  e.  a )  ->  E. v  e.  (metUnif `  D )
( v " {
x } )  C_  a )
2825, 27r19.29a 2996 . . . . . 6  |-  ( ( ( ( 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 ) )
2928ralrimiva 2871 . . . . 5  |-  ( ( ( 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
) )
3012, 29jca 532 . . . 4  |-  ( ( ( 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 5867 . . . . . . 7  |-  ( ball `  D )  e.  _V
3231rnex 6708 . . . . . 6  |-  ran  ( ball `  D )  e. 
_V
33 eltg2 19219 . . . . . 6  |-  ( 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 ) ) ) )
3432, 33ax-mp 5 . . . . 5  |-  ( 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 ) ) )
3534a1i 11 . . . 4  |-  ( ( ( 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 ) ) ) )
3630, 35mpbird 232 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ( topGen ` 
ran  ( ball `  D
) ) )
3734a1i 11 . . . . . . . . 9  |-  ( ( 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 ) ) ) )
3837biimpa 484 . . . . . . . 8  |-  ( ( ( 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 ) ) )
3938simpld 459 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  U. ran  ( ball `  D )
)
4010ad2antlr 726 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  U. ran  ( ball `  D )  =  X )
4139, 40sseqtrd 3533 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  X
)
42 elpwg 4011 . . . . . . 7  |-  ( a  e.  ( topGen `  ran  ( ball `  D )
)  ->  ( a  e.  ~P X  <->  a  C_  X ) )
4342adantl 466 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  ( a  e. 
~P X  <->  a  C_  X ) )
4441, 43mpbird 232 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  ~P X )
45 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  D  e.  (PsMet `  X ) )
4641sselda 3497 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  x  e.  X )
4738simprd 463 . . . . . . . . . . 11  |-  ( ( ( 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
) )
4847r19.21bi 2826 . . . . . . . . . 10  |-  ( ( ( ( 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
) )
49 blssexps 20657 . . . . . . . . . . 11  |-  ( ( 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 ) )
5045, 46, 49syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( 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
) )
5148, 50mpbid 210 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. d  e.  RR+  ( x ( ball `  D ) d ) 
C_  a )
52 blval2 20806 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
53523expa 1191 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5453sseq1d 3524 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
5554rexbidva 2963 . . . . . . . . . 10  |-  ( ( 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 ) )
5655biimpa 484 . . . . . . . . 9  |-  ( ( ( 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 )
5745, 46, 51, 56syl21anc 1222 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a )
58 cnvexg 6720 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
59 imaexg 6711 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6058, 59syl 16 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) d ) )  e. 
_V )
6160ralrimivw 2872 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
62 eqid 2460 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
63 imaeq1 5323 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
6463sseq1d 3524 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
6562, 64rexrnmpt 6022 . . . . . . . . 9  |-  ( 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 )
)
6645, 61, 653syl 20 . . . . . . . 8  |-  ( ( ( ( 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 ) )
6757, 66mpbird 232 . . . . . . 7  |-  ( ( ( ( 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 )
68 oveq2 6283 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
6968imaeq2d 5328 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
7069cbvmptv 4531 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7170rneqi 5220 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7271metustfbas 20797 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
73 ssfg 20101 . . . . . . . . . . 11  |-  ( 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 ) ) ) ) )
7472, 73syl 16 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  ( ( X  X.  X ) filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ) )
75 metuval 20781 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7675adantl 466 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7774, 76sseqtr4d 3534 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnif `  D
) )
78 ssrexv 3558 . . . . . . . . 9  |-  ( 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 ) )
7977, 78syl 16 . . . . . . . 8  |-  ( ( 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 ) )
8079ad2antrr 725 . . . . . . 7  |-  ( ( ( ( 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 )
)
8167, 80mpd 15 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a )
8281ralrimiva 2871 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  A. x  e.  a  E. v  e.  (metUnif `  D ) ( v
" { x }
)  C_  a )
8344, 82jca 532 . . . 4  |-  ( ( ( 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 ) )
846biimpar 485 . . . 4  |-  ( ( ( 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 )
) )
8583, 84syldan 470 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  (unifTop `  (metUnif `  D )
) )
8636, 85impbida 829 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  (unifTop `  (metUnif `  D
) )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
8786eqrdv 2457 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (unifTop `  (metUnif `  D ) )  =  ( topGen `  ran  ( ball `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3106    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   U.cuni 4238    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991   ran crn 4993   "cima 4995   ` cfv 5579  (class class class)co 6275   0cc0 9481   RR+crp 11209   [,)cico 11520   topGenctg 14682  PsMetcpsmet 18166   ballcbl 18169   fBascfbas 18170   filGencfg 18171  metUnifcmetu 18174  UnifOncust 20430  unifTopcutop 20461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ico 11524  df-topgen 14688  df-psmet 18175  df-bl 18178  df-fbas 18180  df-fg 18181  df-metu 18183  df-fil 20075  df-ust 20431  df-utop 20462
This theorem is referenced by:  xmetutop  20815
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