Theorem metustel 21221

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 2532	. 2 |- ( B e. F <-> B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) )
3		elex 3115	. . . 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 6719	. . . . 5 |- ( D e. (PsMet ` X ) -> `' D e. _V )
6		imaexg 6710	. . . . 5 |- ( `' D e. _V -> ( `' D " ( 0 [,) a ) ) e. _V )
7		eleq1a 2537	. . . . 5 |- ( ( `' D " ( 0 [,) a ) ) e. _V -> ( B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
8	5, 6, 7	3syl 20	. . . 4 |- ( D e. (PsMet ` X ) -> ( B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
9	8	rexlimdvw 2949	. . 3 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
10		eqid 2454	. . . . 5 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
11	10	elrnmpt 5238	. . . 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 352	. 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 257	1 |- ( D e. (PsMet ` X ) -> ( B e. F <-> E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1398   e. wcel 1823   E.wrex 2805   _Vcvv 3106   |-> cmpt 4497   `'ccnv 4987   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   0cc0 9481   RR+crp 11221   [,)cico 11534  PsMetcpsmet 18597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001

This theorem is referenced by:  metustto  21227  metustid  21229  metustexhalf  21233  metustfbas  21235  cfilucfil  21239  metucn  21258