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Theorem psmetutop 21255
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 21244 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
2 utopval 20904 . . . . . . . . . . . 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 2524 . . . . . . . . . 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 482 . . . . . . . 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 457 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ~P X
)
98elpwid 4009 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  X )
10 unirnblps 21091 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
1110ad2antlr 724 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  U. ran  ( ball `  D
)  =  X )
129, 11sseqtr4d 3526 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  U. ran  ( ball `  D ) )
13 simpr 459 . . . . . . . 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 753 . . . . . . . . 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 727 . . . . . . . . . 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 3490 . . . . . . . . 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 21253 . . . . . . . . 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 1226 . . . . . . . 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 3497 . . . . . . . . . . 11  |-  ( ( b  C_  ( v " { x } )  /\  ( v " { x } ) 
C_  a )  -> 
b  C_  a )
2221expcom 433 . . . . . . . . . 10  |-  ( ( v " { x } )  C_  a  ->  ( b  C_  (
v " { x } )  ->  b  C_  a ) )
2322anim2d 563 . . . . . . . . 9  |-  ( ( v " { x } )  C_  a  ->  ( ( x  e.  b  /\  b  C_  ( v " {
x } ) )  ->  ( x  e.  b  /\  b  C_  a ) ) )
2423reximdv 2928 . . . . . . . 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 461 . . . . . . . 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 2823 . . . . . . 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 2868 . . . . 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 530 . . . 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 5858 . . . . . 6  |-  ( ball `  D )  e.  _V
3231rnex 6707 . . . . 5  |-  ran  ( ball `  D )  e. 
_V
33 eltg2 19629 . . . . 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 12 . . . 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 232 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ( topGen ` 
ran  ( ball `  D
) ) )
3632, 33mp1i 12 . . . . . . . . 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 482 . . . . . . . 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 457 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  U. ran  ( ball `  D )
)
3910ad2antlr 724 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  U. ran  ( ball `  D )  =  X )
4038, 39sseqtrd 3525 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  X
)
41 elpwg 4007 . . . . . . 7  |-  ( a  e.  ( topGen `  ran  ( ball `  D )
)  ->  ( a  e.  ~P X  <->  a  C_  X ) )
4241adantl 464 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  ( a  e. 
~P X  <->  a  C_  X ) )
4340, 42mpbird 232 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  ~P X )
44 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  D  e.  (PsMet `  X ) )
4540sselda 3489 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  x  e.  X )
4637simprd 461 . . . . . . . . . . 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 2823 . . . . . . . . . 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 21098 . . . . . . . . . . 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 659 . . . . . . . . . 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 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 )
51 blval2 21247 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
52513expa 1194 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5352sseq1d 3516 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
5453rexbidva 2962 . . . . . . . . . 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 482 . . . . . . . . 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 1225 . . . . . . . 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 6719 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
58 imaexg 6710 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
5957, 58syl 16 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) d ) )  e. 
_V )
6059ralrimivw 2869 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
61 eqid 2454 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
62 imaeq1 5320 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
6362sseq1d 3516 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
6461, 63rexrnmpt 6017 . . . . . . . . 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 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 ) )
6656, 65mpbird 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 )
67 oveq2 6278 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
6867imaeq2d 5325 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
6968cbvmptv 4530 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7069rneqi 5218 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7170metustfbas 21238 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
72 ssfg 20542 . . . . . . . . . . 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 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 ) ) ) ) )
74 metuval 21222 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7574adantl 464 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7673, 75sseqtr4d 3526 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnif `  D
) )
77 ssrexv 3551 . . . . . . . . 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 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 ) )
7978ad2antrr 723 . . . . . . 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 2868 . . . . 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 530 . . . 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 483 . . . 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 468 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  (unifTop `  (metUnif `  D )
) )
8535, 84impbida 830 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  (unifTop `  (metUnif `  D
) )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
8685eqrdv 2451 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   U.cuni 4235    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   0cc0 9481   RR+crp 11221   [,)cico 11534   topGenctg 14930  PsMetcpsmet 18600   ballcbl 18603   fBascfbas 18604   filGencfg 18605  metUnifcmetu 18608  UnifOncust 20871  unifTopcutop 20902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-topgen 14936  df-psmet 18609  df-bl 18612  df-fbas 18614  df-fg 18615  df-metu 18617  df-fil 20516  df-ust 20872  df-utop 20903
This theorem is referenced by:  xmetutop  21256
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