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Theorem cfilucfil2 21576
Description: Given a metric  D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 22235. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
cfilucfil2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( C  e.  (CauFilu `  (metUnif `  D
) )  <->  ( C  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
Distinct variable groups:    x, y, C    x, D, y    x, X, y

Proof of Theorem cfilucfil2
Dummy variables  b 
a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metuval 21564 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
21adantl 468 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
32fveq2d 5869 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  (CauFilu `  (metUnif `  D ) )  =  (CauFilu `  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) ) ) )
43eleq2d 2514 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( C  e.  (CauFilu `  (metUnif `  D
) )  <->  C  e.  (CauFilu `  ( ( X  X.  X ) filGen ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) ) ) ) )
5 oveq2 6298 . . . . . 6  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
65imaeq2d 5168 . . . . 5  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
76cbvmptv 4495 . . . 4  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
87rneqi 5061 . . 3  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
98cfilucfil 21574 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( C  e.  (CauFilu `  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) ) )  <->  ( C  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
104, 9bitrd 257 1  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( C  e.  (CauFilu `  (metUnif `  D
) )  <->  ( C  e.  ( fBas `  X
)  /\  A. x  e.  RR+  E. y  e.  C  ( D "
( y  X.  y
) )  C_  (
0 [,) x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    C_ wss 3404   (/)c0 3731    |-> cmpt 4461    X. cxp 4832   `'ccnv 4833   ran crn 4835   "cima 4837   ` cfv 5582  (class class class)co 6290   0cc0 9539   RR+crp 11302   [,)cico 11637  PsMetcpsmet 18954   fBascfbas 18958   filGencfg 18959  metUnifcmetu 18961  CauFiluccfilu 21301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-2 10668  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-psmet 18962  df-fbas 18967  df-fg 18968  df-metu 18969  df-fil 20861  df-ust 21215  df-cfilu 21302
This theorem is referenced by:  cfilucfil3  22288  cmetcusp  22321
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