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Theorem metustel 20245
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 2529 . 2  |-  ( B  e.  F  <->  B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
3 elex 3079 . . . 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 6626 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  `' D  e.  _V )
6 imaexg 6617 . . . . 5  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
7 eleq1a 2534 . . . . 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 2942 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( E. a  e.  RR+  B  =  ( `' D "
( 0 [,) a
) )  ->  B  e.  _V ) )
10 eqid 2451 . . . . 5  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1110elrnmpt 5186 . . . 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 354 . 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 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3070    |-> cmpt 4450   `'ccnv 4939   ran crn 4941   "cima 4943   ` cfv 5518  (class class class)co 6192   0cc0 9385   RR+crp 11094   [,)cico 11405  PsMetcpsmet 17911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953
This theorem is referenced by:  metustto  20251  metustid  20253  metustexhalf  20257  metustfbas  20259  cfilucfil  20263  metucn  20282
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