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Theorem metustel 21576
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 2522 . 2  |-  ( B  e.  F  <->  B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
3 elex 3022 . . . 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 6727 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
6 imaexg 6718 . . . . 5  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
7 eleq1a 2525 . . . . 5  |-  ( ( `' D " ( 0 [,) a ) )  e.  _V  ->  ( B  =  ( `' D " ( 0 [,) a ) )  ->  B  e.  _V )
)
85, 6, 73syl 18 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( B  =  ( `' D " ( 0 [,) a
) )  ->  B  e.  _V ) )
98rexlimdvw 2855 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( E. a  e.  RR+  B  =  ( `' D "
( 0 [,) a
) )  ->  B  e.  _V ) )
10 eqid 2452 . . . . 5  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1110elrnmpt 5059 . . . 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 360 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a ) ) ) )
142, 13syl5bb 265 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 189    = wceq 1448    e. wcel 1891   E.wrex 2738   _Vcvv 3013    |-> cmpt 4433   `'ccnv 4811   ran crn 4813   "cima 4815   ` cfv 5561  (class class class)co 6276   0cc0 9526   RR+crp 11292   [,)cico 11627  PsMetcpsmet 18965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-xp 4818  df-rel 4819  df-cnv 4820  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825
This theorem is referenced by:  metustto  21579  metustid  21580  metustexhalf  21582  metustfbas  21583  cfilucfil  21585  metucn  21597
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