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Theorem metustelOLD 20786
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 2545 . 2  |-  ( B  e.  F  <->  B  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
3 elex 3122 . . . 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 6727 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  `' D  e.  _V )
6 imaexg 6718 . . . . 5  |-  ( `' D  e.  _V  ->  ( `' D " ( 0 [,) a ) )  e.  _V )
7 eleq1a 2550 . . . . 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 2958 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( E. a  e.  RR+  B  =  ( `' D "
( 0 [,) a
) )  ->  B  e.  _V ) )
10 eqid 2467 . . . . 5  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )
1110elrnmpt 5247 . . . 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 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002   ` cfv 5586  (class class class)co 6282   0cc0 9488   RR+crp 11216   [,)cico 11527   *Metcxmt 18171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  metusttoOLD  20792  metustidOLD  20794  metustexhalfOLD  20798  metustfbasOLD  20800  cfilucfilOLD  20804  metucnOLD  20823
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