MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metutopOLD Structured version   Unicode version

Theorem metutopOLD 20953
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 20942 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  e.  (UnifOn `  X ) )
2 utopval 20603 . . . . . . . . . . . 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 2537 . . . . . . . . . 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 3043 . . . . . . . . . 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 4026 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  C_  X )
10 unirnbl 20791 . . . . . . 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 3546 . . . . 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 3510 . . . . . . . . 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 20951 . . . . . . . . 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 1228 . . . . . . . 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 3517 . . . . . . . . . . 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 2941 . . . . . . . 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 2836 . . . . . . 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 3008 . . . . . 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 2881 . . . . 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 2467 . . . . . . . . 9  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
3231mopnval 20809 . . . . . . . 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 2537 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (
MetOpen `  D )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
35 fvex 5882 . . . . . . . 8  |-  ( ball `  D )  e.  _V
3635rnex 6729 . . . . . . 7  |-  ran  ( ball `  D )  e. 
_V
37 eltg2 19328 . . . . . . 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 3545 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
a  C_  X )
46 elpwg 4024 . . . . . . 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 3509 . . . . . . . . 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 2836 . . . . . . . . . 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 20798 . . . . . . . . . . 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 20719 . . . . . . . . . . . . . 14  |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X )
)
57 blval2 20946 . . . . . . . . . . . . . 14  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5856, 57syl3an1 1261 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  ( x ( ball `  D ) d )  =  ( ( `' D " ( 0 [,) d ) )
" { x }
) )
59583expa 1196 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( *Met `  X
)  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
6059sseq1d 3536 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( *Met `  X
)  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
6160rexbidva 2975 . . . . . . . . . 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 1227 . . . . . . . 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 6741 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
65 imaexg 6732 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6664, 65syl 16 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6766ralrimivw 2882 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
68 eqid 2467 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
69 imaeq1 5338 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
7069sseq1d 3536 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
7168, 70rexrnmpt 6042 . . . . . . . . 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 6303 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
7574imaeq2d 5343 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
7675cbvmptv 4544 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7776rneqi 5235 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7877metustfbasOLD 20936 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
79 ssfg 20241 . . . . . . . . . . 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 20920 . . . . . . . . . . 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 3546 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnifOLD
`  D ) )
84 ssrexv 3570 . . . . . . . . 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 2881 . . . . 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 830 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (unifTop `  (metUnifOLD
`  D ) )  <-> 
a  e.  ( MetOpen `  D ) ) )
9392eqrdv 2464 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   U.cuni 4251    |-> cmpt 4511    X. cxp 5003   `'ccnv 5004   ran crn 5006   "cima 5008   ` cfv 5594  (class class class)co 6295   0cc0 9504   RR+crp 11232   [,)cico 11543   topGenctg 14710  PsMetcpsmet 18272   *Metcxmt 18273   ballcbl 18275   fBascfbas 18276   filGencfg 18277   MetOpencmopn 18278  metUnifOLDcmetuOLD 18279  UnifOncust 20570  unifTopcutop 20601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ico 11547  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-metuOLD 18288  df-fil 20215  df-ust 20571  df-utop 20602
This theorem is referenced by:  xmsuspOLD  20956  cmetcuspOLD  21661
  Copyright terms: Public domain W3C validator