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Theorem metustelOLD 20131
Description: Define a filter base  F generated by a metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustelOLD  |-  ( D  e.  ( *Met `  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 metustelOLD
StepHypRef Expression
1 metust.1 . . 3  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
21eleq2i 2507 . 2  |-  ( B  e.  F  <->  B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
3 elex 2986 . . . 4  |-  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  ->  B  e.  _V )
43a1i 11 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  ->  B  e.  _V ) )
5 cnvexg 6529 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
6 imaexg 6520 . . . . 5  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
7 eleq1a 2512 . . . . 5  |-  ( ( `' D " ( 0 [,) a ) )  e.  _V  ->  ( B  =  ( `' D " ( 0 [,) a ) )  ->  B  e.  _V )
)
85, 6, 73syl 20 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( B  =  ( `' D " ( 0 [,) a ) )  ->  B  e.  _V )
)
98rexlimdvw 2849 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( E. a  e.  RR+  B  =  ( `' D "
( 0 [,) a
) )  ->  B  e.  _V ) )
10 eqid 2443 . . . . 5  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1110elrnmpt 5091 . . . 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.  ( *Met `  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.  ( *Met `  X )  ->  ( B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a
) ) ) )
142, 13syl5bb 257 1  |-  ( D  e.  ( *Met `  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 1369    e. wcel 1756   E.wrex 2721   _Vcvv 2977    e. cmpt 4355   `'ccnv 4844   ran crn 4846   "cima 4848   ` cfv 5423  (class class class)co 6096   0cc0 9287   RR+crp 10996   [,)cico 11307   *Metcxmt 17806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
This theorem is referenced by:  metusttoOLD  20137  metustidOLD  20139  metustexhalfOLD  20143  metustfbasOLD  20145  cfilucfilOLD  20149  metucnOLD  20168
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