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Theorem metutopOLD 20162
Description: The topology induced by a uniform structure generated by a metric  D is that metric's open sets. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
metutopOLD  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(unifTop `  (metUnifOLD
`  D ) )  =  ( MetOpen `  D
) )

Proof of Theorem metutopOLD
Dummy variables  a 
b  d  e  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metuustOLD 20151 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  e.  (UnifOn `  X ) )
2 utopval 19812 . . . . . . . . . . . 12  |-  ( (metUnifOLD `  D )  e.  (UnifOn `  X )  ->  (unifTop `  (metUnifOLD
`  D ) )  =  { a  e. 
~P X  |  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a } )
31, 2syl 16 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(unifTop `  (metUnifOLD
`  D ) )  =  { a  e. 
~P X  |  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a } )
43eleq2d 2510 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (unifTop `  (metUnifOLD
`  D ) )  <-> 
a  e.  { a  e.  ~P X  |  A. x  e.  a  E. v  e.  (metUnifOLD `  D
) ( v " { x } ) 
C_  a } ) )
5 rabid 2902 . . . . . . . . . 10  |-  ( a  e.  { a  e. 
~P X  |  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a } 
<->  ( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
) )
64, 5syl6bb 261 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (unifTop `  (metUnifOLD
`  D ) )  <-> 
( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
) ) )
76biimpa 484 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
) )
87simpld 459 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  e.  ~P X
)
98elpwid 3875 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  C_  X )
10 unirnbl 20000 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  =  X )
1110ad2antlr 726 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  ->  U. ran  ( ball `  D
)  =  X )
129, 11sseqtr4d 3398 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  C_  U. ran  ( ball `  D ) )
13 simpr 461 . . . . . . . 8  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  ( v " { x } ) 
C_  a )
14 simp-5r 768 . . . . . . . . 9  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  D  e.  ( *Met `  X
) )
15 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  v  e.  (metUnifOLD `  D ) )
169ad3antrrr 729 . . . . . . . . . 10  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  a  C_  X )
17 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  x  e.  a )
1816, 17sseldd 3362 . . . . . . . . 9  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  x  e.  X )
19 metustblOLD 20160 . . . . . . . . 9  |-  ( ( D  e.  ( *Met `  X )  /\  v  e.  (metUnifOLD `  D )  /\  x  e.  X )  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  ( v " {
x } ) ) )
2014, 15, 18, 19syl3anc 1218 . . . . . . . 8  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  ( v " {
x } ) ) )
21 sstr 3369 . . . . . . . . . . 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 2832 . . . . . . . 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.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  /\  x  e.  a )  /\  v  e.  (metUnifOLD `  D
) )  /\  (
v " { x } )  C_  a
)  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a ) )
267simprd 463 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  ->  A. x  e.  a  E. v  e.  (metUnifOLD `  D
) ( v " { x } ) 
C_  a )
2726r19.21bi 2819 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  (unifTop `  (metUnifOLD
`  D ) ) )  /\  x  e.  a )  ->  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
)
2825, 27r19.29a 2867 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  (unifTop `  (metUnifOLD
`  D ) ) )  /\  x  e.  a )  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a ) )
2928ralrimiva 2804 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  ->  A. x  e.  a  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) )
3012, 29jca 532 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
( a  C_  U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) ) )
31 eqid 2443 . . . . . . . . 9  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
3231mopnval 20018 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  D )  =  ( topGen `  ran  ( ball `  D ) ) )
3332adantl 466 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( MetOpen `  D )  =  ( topGen `  ran  ( ball `  D )
) )
3433eleq2d 2510 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (
MetOpen `  D )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
35 fvex 5706 . . . . . . . 8  |-  ( ball `  D )  e.  _V
3635rnex 6517 . . . . . . 7  |-  ran  ( ball `  D )  e. 
_V
37 eltg2 18568 . . . . . . 7  |-  ( 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 ) ) ) )
3836, 37ax-mp 5 . . . . . 6  |-  ( 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 ) ) )
3934, 38syl6bb 261 . . . . 5  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (
MetOpen `  D )  <->  ( a  C_ 
U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D ) ( x  e.  b  /\  b  C_  a ) ) ) )
4039adantr 465 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
( a  e.  (
MetOpen `  D )  <->  ( a  C_ 
U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D ) ( x  e.  b  /\  b  C_  a ) ) ) )
4130, 40mpbird 232 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  e.  ( MetOpen `  D ) )
4239biimpa 484 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
( a  C_  U. ran  ( ball `  D )  /\  A. x  e.  a  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) ) )
4342simpld 459 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
a  C_  U. ran  ( ball `  D ) )
4410ad2antlr 726 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  ->  U. ran  ( ball `  D
)  =  X )
4543, 44sseqtrd 3397 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
a  C_  X )
46 elpwg 3873 . . . . . . 7  |-  ( a  e.  ( MetOpen `  D
)  ->  ( a  e.  ~P X  <->  a  C_  X ) )
4746adantl 466 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
( a  e.  ~P X 
<->  a  C_  X )
)
4845, 47mpbird 232 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
a  e.  ~P X
)
49 simpllr 758 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  D  e.  ( *Met `  X ) )
5045sselda 3361 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  x  e.  X )
5142simprd 463 . . . . . . . . . . 11  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  ->  A. x  e.  a  E. b  e.  ran  ( ball `  D )
( x  e.  b  /\  b  C_  a
) )
5251r19.21bi 2819 . . . . . . . . . 10  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  E. b  e.  ran  ( ball `  D
) ( x  e.  b  /\  b  C_  a ) )
53 blssex 20007 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  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
) )
5449, 50, 53syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  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 ) )
5552, 54mpbid 210 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  E. d  e.  RR+  ( x (
ball `  D )
d )  C_  a
)
56 xmetpsmet 19928 . . . . . . . . . . . . . 14  |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X )
)
57 blval2 20155 . . . . . . . . . . . . . 14  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5856, 57syl3an1 1251 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  ( x ( ball `  D ) d )  =  ( ( `' D " ( 0 [,) d ) )
" { x }
) )
59583expa 1187 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( *Met `  X
)  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
6059sseq1d 3388 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( *Met `  X
)  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
6160rexbidva 2737 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X
)  ->  ( E. d  e.  RR+  ( x ( ball `  D
) d )  C_  a 
<->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
6261biimpa 484 . . . . . . . . 9  |-  ( ( ( D  e.  ( *Met `  X
)  /\  x  e.  X )  /\  E. d  e.  RR+  ( x ( ball `  D
) d )  C_  a )  ->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
6349, 50, 55, 62syl21anc 1217 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  E. d  e.  RR+  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
64 cnvexg 6529 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
65 imaexg 6520 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6664, 65syl 16 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6766ralrimivw 2805 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
68 eqid 2443 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
69 imaeq1 5169 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
7069sseq1d 3388 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
7168, 70rexrnmpt 5858 . . . . . . . . 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 )
)
7249, 67, 713syl 20 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  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 )
)
7363, 72mpbird 232 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  E. v  e.  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ( v " { x } )  C_  a
)
74 oveq2 6104 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
7574imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
7675cbvmptv 4388 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7776rneqi 5071 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7877metustfbasOLD 20145 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
79 ssfg 19450 . . . . . . . . . . 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 ) ) ) ) )
8078, 79syl 16 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  ( ( X  X.  X ) filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ) )
81 metuvalOLD 20129 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  (metUnifOLD `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
8281adantl 466 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  =  ( ( X  X.  X
) filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
8380, 82sseqtr4d 3398 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnifOLD
`  D ) )
84 ssrexv 3422 . . . . . . . . 9  |-  ( ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnifOLD
`  D )  -> 
( E. v  e. 
ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ( v " { x } )  C_  a  ->  E. v  e.  (metUnifOLD `  D ) ( v
" { x }
)  C_  a )
)
8583, 84syl 16 . . . . . . . 8  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( E. v  e. 
ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ( v " { x } )  C_  a  ->  E. v  e.  (metUnifOLD `  D ) ( v
" { x }
)  C_  a )
)
8685ad2antrr 725 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  ( E. v  e.  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) 
C_  a  ->  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
) )
8773, 86mpd 15 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
)
8887ralrimiva 2804 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  ->  A. x  e.  a  E. v  e.  (metUnifOLD `  D
) ( v " { x } ) 
C_  a )
8948, 88jca 532 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
) )
906biimpar 485 . . . 4  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( a  e.  ~P X  /\  A. x  e.  a  E. v  e.  (metUnifOLD
`  D ) ( v " { x } )  C_  a
) )  ->  a  e.  (unifTop `  (metUnifOLD
`  D ) ) )
9189, 90syldan 470 . . 3  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
a  e.  (unifTop `  (metUnifOLD `  D
) ) )
9241, 91impbida 828 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (unifTop `  (metUnifOLD
`  D ) )  <-> 
a  e.  ( MetOpen `  D ) ) )
9392eqrdv 2441 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(unifTop `  (metUnifOLD
`  D ) )  =  ( MetOpen `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   {csn 3882   U.cuni 4096    e. cmpt 4355    X. cxp 4843   `'ccnv 4844   ran crn 4846   "cima 4848   ` cfv 5423  (class class class)co 6096   0cc0 9287   RR+crp 10996   [,)cico 11307   topGenctg 14381  PsMetcpsmet 17805   *Metcxmt 17806   ballcbl 17808   fBascfbas 17809   filGencfg 17810   MetOpencmopn 17811  metUnifOLDcmetuOLD 17812  UnifOncust 19779  unifTopcutop 19810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ico 11311  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-metuOLD 17821  df-fil 19424  df-ust 19780  df-utop 19811
This theorem is referenced by:  xmsuspOLD  20165  cmetcuspOLD  20870
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