Theorem metustel 20245

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 2529	. 2 |- ( B e. F <-> B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) )
3		elex 3079	. . . 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 6626	. . . . 5 |- ( D e. (PsMet ` X ) -> `' D e. _V )
6		imaexg 6617	. . . . 5 |- ( `' D e. _V -> ( `' D " ( 0 [,) a ) ) e. _V )
7		eleq1a 2534	. . . . 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 2942	. . 3 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
10		eqid 2451	. . . . 5 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
11	10	elrnmpt 5186	. . . 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 1370   e. wcel 1758   E.wrex 2796   _Vcvv 3070   |-> cmpt 4450   `'ccnv 4939   ran crn 4941   "cima 4943   ` cfv 5518  (class class class)co 6192   0cc0 9385   RR+crp 11094   [,)cico 11405  PsMetcpsmet 17911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953

This theorem is referenced by:  metustto  20251  metustid  20253  metustexhalf  20257  metustfbas  20259  cfilucfil  20263  metucn  20282