Theorem metustel 20790

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 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 )
4	3	a1i 11	. . 3 |- ( D e. (PsMet ` X ) -> ( B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) -> B e. _V ) )
5		cnvexg 6727	. . . . 5 |- ( D e. (PsMet ` 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 ) )
8	5, 6, 7	3syl 20	. . . 4 |- ( D e. (PsMet ` X ) -> ( B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
9	8	rexlimdvw 2958	. . 3 |- ( D e. (PsMet ` 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 ) ) )
11	10	elrnmpt 5247	. . . 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 354	. 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 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  PsMetcpsmet 18173

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:  metustto  20796  metustid  20798  metustexhalf  20802  metustfbas  20804  cfilucfil  20808  metucn  20827