Theorem metustel 22355

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

Step	Hyp	Ref	Expression
1		metust.1	. . 3 |- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
2	1	eleq2i 2693	. 2 |- ( B e. F <-> B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) )
3		elex 3212	. . . 4 |- ( B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) -> B e. _V )
4	3	a1i 11	. . 3 |- ( D e. (PsMet ` X ) -> ( B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) -> B e. _V ) )
5		cnvexg 7112	. . . . 5 |- ( D e. (PsMet ` X ) -> `' D e. _V )
6		imaexg 7103	. . . . 5 |- ( `' D e. _V -> ( `' D " ( 0 [,) a ) ) e. _V )
7		eleq1a 2696	. . . . 5 |- ( ( `' D " ( 0 [,) a ) ) e. _V -> ( B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
8	5, 6, 7	3syl 18	. . . 4 |- ( D e. (PsMet ` X ) -> ( B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
9	8	rexlimdvw 3034	. . 3 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
10		eqid 2622	. . . . 5 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
11	10	elrnmpt 5372	. . . 4 |- ( B e. _V -> ( B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) ) )
12	11	a1i 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 ) ) ) ) )
13	4, 9, 12	pm5.21ndd 369	. 2 |- ( D e. (PsMet ` X ) -> ( B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) ) )
14	2, 13	syl5bb 272	1 |- ( D e. (PsMet ` X ) -> ( B e. F <-> E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  metustto  22358  metustid  22359  metustexhalf  22361  metustfbas  22362  cfilucfil  22364  metucn  22376