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Theorem psmetutop 21660
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 21653 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  e.  (UnifOn `  X ) )
2 utopval 21325 . . . . . . . . . . . 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 2534 . . . . . . . . . 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 2953 . . . . . . . . . 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 269 . . . . . . . . 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 492 . . . . . . . 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 466 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  e.  ~P X
)
98elpwid 3952 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  X )
10 unirnblps 21512 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  U. ran  ( ball `  D )  =  X )
1110ad2antlr 741 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  ->  U. ran  ( ball `  D
)  =  X )
129, 11sseqtr4d 3455 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  (unifTop `  (metUnif `  D
) ) )  -> 
a  C_  U. ran  ( ball `  D ) )
13 simpr 468 . . . . . . . 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 787 . . . . . . . . 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 770 . . . . . . . . 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 744 . . . . . . . . . 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 777 . . . . . . . . . 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 3419 . . . . . . . . 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 21659 . . . . . . . . 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 1292 . . . . . . . 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 3426 . . . . . . . . . . 11  |-  ( ( b  C_  ( v " { x } )  /\  ( v " { x } ) 
C_  a )  -> 
b  C_  a )
2221expcom 442 . . . . . . . . . 10  |-  ( ( v " { x } )  C_  a  ->  ( b  C_  (
v " { x } )  ->  b  C_  a ) )
2322anim2d 575 . . . . . . . . 9  |-  ( ( v " { x } )  C_  a  ->  ( ( x  e.  b  /\  b  C_  ( v " {
x } ) )  ->  ( x  e.  b  /\  b  C_  a ) ) )
2423reximdv 2857 . . . . . . . 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 61 . . . . . . 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 470 . . . . . . . 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 2776 . . . . . . 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 2918 . . . . . 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 2809 . . . . 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 541 . . . 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 5889 . . . . . 6  |-  ( ball `  D )  e.  _V
3231rnex 6746 . . . . 5  |-  ran  ( ball `  D )  e. 
_V
33 eltg2 20050 . . . . 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 240 . . 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 492 . . . . . . . 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 466 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  U. ran  ( ball `  D )
)
3910ad2antlr 741 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  U. ran  ( ball `  D )  =  X )
4038, 39sseqtrd 3454 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  C_  X
)
41 elpwg 3950 . . . . . . 7  |-  ( a  e.  ( topGen `  ran  ( ball `  D )
)  ->  ( a  e.  ~P X  <->  a  C_  X ) )
4241adantl 473 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  ( a  e. 
~P X  <->  a  C_  X ) )
4340, 42mpbird 240 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  ~P X )
44 simpllr 777 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  D  e.  (PsMet `  X ) )
4540sselda 3418 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  a  e.  ( topGen ` 
ran  ( ball `  D
) ) )  /\  x  e.  a )  ->  x  e.  X )
4637simprd 470 . . . . . . . . . . 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 2776 . . . . . . . . . 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 21519 . . . . . . . . . . 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 673 . . . . . . . . . 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 215 . . . . . . . . 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 21655 . . . . . . . . . . . . 13  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
52513expa 1231 . . . . . . . . . . . 12  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5352sseq1d 3445 . . . . . . . . . . 11  |-  ( ( ( D  e.  (PsMet `  X )  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
5453rexbidva 2889 . . . . . . . . . 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 492 . . . . . . . . 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 1291 . . . . . . . 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 6758 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
58 imaexg 6749 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
5957, 58syl 17 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) d ) )  e. 
_V )
6059ralrimivw 2810 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
61 eqid 2471 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
62 imaeq1 5169 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
6362sseq1d 3445 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
6461, 63rexrnmpt 6047 . . . . . . . . 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 240 . . . . . . 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 6316 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
6867imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
6968cbvmptv 4488 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7069rneqi 5067 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7170metustfbas 21650 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
72 ssfg 20965 . . . . . . . . . . 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 21642 . . . . . . . . . . 11  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7574adantl 473 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
7673, 75sseqtr4d 3455 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnif `  D
) )
77 ssrexv 3480 . . . . . . . . 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 740 . . . . . . 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 2809 . . . . 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 541 . . . 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 493 . . . 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 478 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  a  e.  ( topGen `  ran  ( ball `  D ) ) )  ->  a  e.  (unifTop `  (metUnif `  D )
) )
8535, 84impbida 850 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( a  e.  (unifTop `  (metUnif `  D
) )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
8685eqrdv 2469 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (unifTop `  (metUnif `  D ) )  =  ( topGen `  ran  ( ball `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   U.cuni 4190    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   ran crn 4840   "cima 4842   ` cfv 5589  (class class class)co 6308   0cc0 9557   RR+crp 11325   [,)cico 11662   topGenctg 15414  PsMetcpsmet 19031   ballcbl 19034   fBascfbas 19035   filGencfg 19036  metUnifcmetu 19038  UnifOncust 21292  unifTopcutop 21323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ico 11666  df-topgen 15420  df-psmet 19039  df-bl 19042  df-fbas 19044  df-fg 19045  df-metu 19046  df-fil 20939  df-ust 21293  df-utop 21324
This theorem is referenced by:  xmetutop  21661
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