Theorem metustel 21576

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 2522	. 2 |- ( B e. F <-> B e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) )
3		elex 3022	. . . 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 2525	. . . . 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 2855	. . 3 |- ( D e. (PsMet ` X ) -> ( E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) -> B e. _V ) )
10		eqid 2452	. . . . 5 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
11	10	elrnmpt 5059	. . . 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 360	. 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 265	1 |- ( D e. (PsMet ` X ) -> ( B e. F <-> E. a e. RR+ B = ( `' D " ( 0 [,) a ) ) ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1448   e. wcel 1891   E.wrex 2738   _Vcvv 3013   |-> cmpt 4433   `'ccnv 4811   ran crn 4813   "cima 4815   ` cfv 5561  (class class class)co 6276   0cc0 9526   RR+crp 11292   [,)cico 11627  PsMetcpsmet 18965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-xp 4818  df-rel 4819  df-cnv 4820  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825

This theorem is referenced by:  metustto  21579  metustid  21580  metustexhalf  21582  metustfbas  21583  cfilucfil  21585  metucn  21597