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Theorem psmetutop 21582
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 21575 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
2 utopval 21247 . . . . . . . . . . . 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 17 . . . . . . . . . . 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 2514 . . . . . . . . . 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 2967 . . . . . . . . . 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 265 . . . . . . . . 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 487 . . . . . . . 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 461 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ~P X
)
98elpwid 3961 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  X )
10 unirnblps 21434 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
1110ad2antlr 733 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  U. ran  ( ball `  D
)  =  X )
129, 11sseqtr4d 3469 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  U. ran  ( ball `  D ) )
13 simpr 463 . . . . . . . 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 779 . . . . . . . . 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 762 . . . . . . . . 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 736 . . . . . . . . . 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 769 . . . . . . . . . 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 3433 . . . . . . . . 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 21581 . . . . . . . . 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 1268 . . . . . . . 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 3440 . . . . . . . . . . 11  |-  ( ( b  C_  ( v " { x } )  /\  ( v " { x } ) 
C_  a )  -> 
b  C_  a )
2221expcom 437 . . . . . . . . . 10  |-  ( ( v " { x } )  C_  a  ->  ( b  C_  (
v " { x } )  ->  b  C_  a ) )
2322anim2d 569 . . . . . . . . 9  |-  ( ( v " { x } )  C_  a  ->  ( ( x  e.  b  /\  b  C_  ( v " {
x } ) )  ->  ( x  e.  b  /\  b  C_  a ) ) )
2423reximdv 2861 . . . . . . . 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 62 . . . . . . 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 465 . . . . . . . 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 2757 . . . . . . 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 2932 . . . . . 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 2802 . . . . 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 535 . . . 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 5875 . . . . . 6  |-  ( ball `  D )  e.  _V
3231rnex 6727 . . . . 5  |-  ran  ( ball `  D )  e. 
_V
33 eltg2 19973 . . . . 5  |-  ( 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, 33mp1i 13 . . . 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 ) ) ) )
3530, 34mpbird 236 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ( topGen ` 
ran  ( ball `  D
) ) )
3632, 33mp1i 13 . . . . . . . . 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 ) ) ) )
3736biimpa 487 . . . . . . . 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 ) ) )
3837simpld 461 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  U. ran  ( ball `  D )
)
3910ad2antlr 733 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  U. ran  ( ball `  D )  =  X )
4038, 39sseqtrd 3468 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  X
)
41 elpwg 3959 . . . . . . 7  |-  ( a  e.  ( topGen `  ran  ( ball `  D )
)  ->  ( a  e.  ~P X  <->  a  C_  X ) )
4241adantl 468 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  ( a  e. 
~P X  <->  a  C_  X ) )
4340, 42mpbird 236 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  ~P X )
44 simpllr 769 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  D  e.  (PsMet `  X ) )
4540sselda 3432 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  x  e.  X )
4637simprd 465 . . . . . . . . . . 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
) )
4746r19.21bi 2757 . . . . . . . . . 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
) )
48 blssexps 21441 . . . . . . . . . . 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 ) )
4944, 45, 48syl2anc 667 . . . . . . . . . 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
) )
5047, 49mpbid 214 . . . . . . . . 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 )
51 blval2 21577 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
52513expa 1208 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5352sseq1d 3459 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
5453rexbidva 2898 . . . . . . . . . 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 ) )
5554biimpa 487 . . . . . . . . 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 )
5644, 45, 50, 55syl21anc 1267 . . . . . . . 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 )
57 cnvexg 6739 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
58 imaexg 6730 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
5957, 58syl 17 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) d ) )  e. 
_V )
6059ralrimivw 2803 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
61 eqid 2451 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
62 imaeq1 5163 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
6362sseq1d 3459 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
6461, 63rexrnmpt 6032 . . . . . . . . 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 )
)
6544, 60, 643syl 18 . . . . . . . 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 ) )
6656, 65mpbird 236 . . . . . . 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 )
67 oveq2 6298 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
6867imaeq2d 5168 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
6968cbvmptv 4495 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7069rneqi 5061 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7170metustfbas 21572 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
72 ssfg 20887 . . . . . . . . . . 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 ) ) ) ) )
7371, 72syl 17 . . . . . . . . . 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 ) ) ) ) )
74 metuval 21564 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7574adantl 468 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7673, 75sseqtr4d 3469 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnif `  D
) )
77 ssrexv 3494 . . . . . . . . 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 ) )
7876, 77syl 17 . . . . . . . 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 ) )
7978ad2antrr 732 . . . . . . 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 )
)
8066, 79mpd 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 )
8180ralrimiva 2802 . . . . 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 )
8243, 81jca 535 . . . 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 ) )
836biimpar 488 . . . 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 )
) )
8482, 83syldan 473 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  (unifTop `  (metUnif `  D )
) )
8535, 84impbida 843 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  (unifTop `  (metUnif `  D
) )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
8685eqrdv 2449 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (unifTop `  (metUnif `  D ) )  =  ( topGen `  ran  ( ball `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {csn 3968   U.cuni 4198    |-> cmpt 4461    X. cxp 4832   `'ccnv 4833   ran crn 4835   "cima 4837   ` cfv 5582  (class class class)co 6290   0cc0 9539   RR+crp 11302   [,)cico 11637   topGenctg 15336  PsMetcpsmet 18954   ballcbl 18957   fBascfbas 18958   filGencfg 18959  metUnifcmetu 18961  UnifOncust 21214  unifTopcutop 21245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-topgen 15342  df-psmet 18962  df-bl 18965  df-fbas 18967  df-fg 18968  df-metu 18969  df-fil 20861  df-ust 21215  df-utop 21246
This theorem is referenced by:  xmetutop  21583
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