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Theorem metustel 21221
Description: Define a filter base  F generated by a metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustel  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  F  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a
) ) ) )
Distinct variable groups:    B, a    D, a    X, a
Allowed substitution hint:    F( a)

Proof of Theorem metustel
StepHypRef Expression
1 metust.1 . . 3  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21eleq2i 2532 . 2  |-  ( B  e.  F  <->  B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
3 elex 3115 . . . 4  |-  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  ->  B  e.  _V )
43a1i 11 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )  ->  B  e.  _V )
)
5 cnvexg 6719 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
6 imaexg 6710 . . . . 5  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
7 eleq1a 2537 . . . . 5  |-  ( ( `' D " ( 0 [,) a ) )  e.  _V  ->  ( B  =  ( `' D " ( 0 [,) a ) )  ->  B  e.  _V )
)
85, 6, 73syl 20 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( B  =  ( `' D " ( 0 [,) a
) )  ->  B  e.  _V ) )
98rexlimdvw 2949 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( E. a  e.  RR+  B  =  ( `' D "
( 0 [,) a
) )  ->  B  e.  _V ) )
10 eqid 2454 . . . . 5  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1110elrnmpt 5238 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a
) ) ) )
1211a1i 11 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  _V  ->  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a ) ) ) ) )
134, 9, 12pm5.21ndd 352 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a ) ) ) )
142, 13syl5bb 257 1  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  F  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106    |-> cmpt 4497   `'ccnv 4987   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   0cc0 9481   RR+crp 11221   [,)cico 11534  PsMetcpsmet 18597
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
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by:  metustto  21227  metustid  21229  metustexhalf  21233  metustfbas  21235  cfilucfil  21239  metucn  21258
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