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Theorem metutopOLD 21170
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 21159 . . . . . . . . . . . 12  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  e.  (UnifOn `  X ) )
2 utopval 20820 . . . . . . . . . . . 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 2452 . . . . . . . . . 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 2959 . . . . . . . . . 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 482 . . . . . . . 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 457 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  e.  ~P X
)
98elpwid 3937 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  C_  X )
10 unirnbl 21008 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  =  X )
1110ad2antlr 724 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  ->  U. ran  ( ball `  D
)  =  X )
129, 11sseqtr4d 3454 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  (unifTop `  (metUnifOLD `  D
) ) )  -> 
a  C_  U. ran  ( ball `  D ) )
13 simpr 459 . . . . . . . 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 753 . . . . . . . . 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 727 . . . . . . . . . 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 3418 . . . . . . . . 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 21168 . . . . . . . . 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 1226 . . . . . . . 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 3425 . . . . . . . . . . 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 2856 . . . . . . . 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 461 . . . . . . . 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 2751 . . . . . . 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 2924 . . . . . 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 2796 . . . . 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 530 . . . 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 2382 . . . . . . . . 9  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
3231mopnval 21026 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  D )  =  ( topGen `  ran  ( ball `  D ) ) )
3332adantl 464 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( MetOpen `  D )  =  ( topGen `  ran  ( ball `  D )
) )
3433eleq2d 2452 . . . . . 6  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( a  e.  (
MetOpen `  D )  <->  a  e.  ( topGen `  ran  ( ball `  D ) ) ) )
35 fvex 5784 . . . . . . . 8  |-  ( ball `  D )  e.  _V
3635rnex 6633 . . . . . . 7  |-  ran  ( ball `  D )  e. 
_V
37 eltg2 19544 . . . . . . 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 463 . . . 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 482 . . . . . . . 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 457 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
a  C_  U. ran  ( ball `  D ) )
4410ad2antlr 724 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  ->  U. ran  ( ball `  D
)  =  X )
4543, 44sseqtrd 3453 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  a  e.  ( MetOpen `  D ) )  -> 
a  C_  X )
46 elpwg 3935 . . . . . . 7  |-  ( a  e.  ( MetOpen `  D
)  ->  ( a  e.  ~P X  <->  a  C_  X ) )
4746adantl 464 . . . . . 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 3417 . . . . . . . . 9  |-  ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X
) )  /\  a  e.  ( MetOpen `  D )
)  /\  x  e.  a )  ->  x  e.  X )
5142simprd 461 . . . . . . . . . . 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 2751 . . . . . . . . . 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 21015 . . . . . . . . . . 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 659 . . . . . . . . . 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 20936 . . . . . . . . . . . . . 14  |-  ( D  e.  ( *Met `  X )  ->  D  e.  (PsMet `  X )
)
57 blval2 21163 . . . . . . . . . . . . . 14  |-  ( ( D  e.  (PsMet `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
5856, 57syl3an1 1259 . . . . . . . . . . . . 13  |-  ( ( D  e.  ( *Met `  X )  /\  x  e.  X  /\  d  e.  RR+ )  ->  ( x ( ball `  D ) d )  =  ( ( `' D " ( 0 [,) d ) )
" { x }
) )
59583expa 1194 . . . . . . . . . . . 12  |-  ( ( ( D  e.  ( *Met `  X
)  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
x ( ball `  D
) d )  =  ( ( `' D " ( 0 [,) d
) ) " {
x } ) )
6059sseq1d 3444 . . . . . . . . . . 11  |-  ( ( ( D  e.  ( *Met `  X
)  /\  x  e.  X )  /\  d  e.  RR+ )  ->  (
( x ( ball `  D ) d ) 
C_  a  <->  ( ( `' D " ( 0 [,) d ) )
" { x }
)  C_  a )
)
6160rexbidva 2890 . . . . . . . . . 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 482 . . . . . . . . 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 1225 . . . . . . . 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 6645 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
65 imaexg 6636 . . . . . . . . . . 11  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6664, 65syl 16 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " ( 0 [,) d ) )  e.  _V )
6766ralrimivw 2797 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  A. d  e.  RR+  ( `' D " ( 0 [,) d
) )  e.  _V )
68 eqid 2382 . . . . . . . . . 10  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )
69 imaeq1 5244 . . . . . . . . . . 11  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
v " { x } )  =  ( ( `' D "
( 0 [,) d
) ) " {
x } ) )
7069sseq1d 3444 . . . . . . . . . 10  |-  ( v  =  ( `' D " ( 0 [,) d
) )  ->  (
( v " {
x } )  C_  a 
<->  ( ( `' D " ( 0 [,) d
) ) " {
x } )  C_  a ) )
7168, 70rexrnmpt 5943 . . . . . . . . 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 6204 . . . . . . . . . . . . . . 15  |-  ( d  =  e  ->  (
0 [,) d )  =  ( 0 [,) e ) )
7574imaeq2d 5249 . . . . . . . . . . . . . 14  |-  ( d  =  e  ->  ( `' D " ( 0 [,) d ) )  =  ( `' D " ( 0 [,) e
) ) )
7675cbvmptv 4458 . . . . . . . . . . . . 13  |-  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ( e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7776rneqi 5142 . . . . . . . . . . . 12  |-  ran  (
d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  =  ran  (
e  e.  RR+  |->  ( `' D " ( 0 [,) e ) ) )
7877metustfbasOLD 21153 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  e.  ( fBas `  ( X  X.  X
) ) )
79 ssfg 20458 . . . . . . . . . . 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 21137 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  (metUnifOLD `  D
)  =  ( ( X  X.  X )
filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
8281adantl 464 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  =  ( ( X  X.  X
) filGen ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d
) ) ) ) )
8380, 82sseqtr4d 3454 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( d  e.  RR+  |->  ( `' D " ( 0 [,) d ) ) )  C_  (metUnifOLD
`  D ) )
84 ssrexv 3479 . . . . . . . . 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 723 . . . . . . 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 2796 . . . . 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 530 . . . 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 483 . . . 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 468 . . 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 2379 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(unifTop `  (metUnifOLD
`  D ) )  =  ( MetOpen `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   {csn 3944   U.cuni 4163    |-> cmpt 4425    X. cxp 4911   `'ccnv 4912   ran crn 4914   "cima 4916   ` cfv 5496  (class class class)co 6196   0cc0 9403   RR+crp 11139   [,)cico 11452   topGenctg 14845  PsMetcpsmet 18515   *Metcxmt 18516   ballcbl 18518   fBascfbas 18519   filGencfg 18520   MetOpencmopn 18521  metUnifOLDcmetuOLD 18522  UnifOncust 20787  unifTopcutop 20818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ico 11456  df-topgen 14851  df-psmet 18524  df-xmet 18525  df-bl 18527  df-mopn 18528  df-fbas 18529  df-fg 18530  df-metuOLD 18531  df-fil 20432  df-ust 20788  df-utop 20819
This theorem is referenced by:  xmsuspOLD  21173  cmetcuspOLD  21878
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