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Theorem cfilucfil2OLD 21201
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 21829. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cfilucfil2OLD  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( C  e.  (CauFilu `  (metUnifOLD
`  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 cfilucfil2OLD
Dummy variables  b 
a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metuvalOLD 21177 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (metUnifOLD `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
21adantl 466 . . . 4  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(metUnifOLD `  D )  =  ( ( X  X.  X
) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
32fveq2d 5876 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
(CauFilu
`  (metUnifOLD
`  D ) )  =  (CauFilu `  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) ) ) )
43eleq2d 2527 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( C  e.  (CauFilu `  (metUnifOLD
`  D ) )  <-> 
C  e.  (CauFilu `  (
( X  X.  X
) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) ) ) )
5 oveq2 6304 . . . . . 6  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
65imaeq2d 5347 . . . . 5  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
76cbvmptv 4548 . . . 4  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
87rneqi 5239 . . 3  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
98cfilucfilOLD 21197 . 2  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  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 253 1  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( C  e.  (CauFilu `  (metUnifOLD
`  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 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    C_ wss 3471   (/)c0 3793    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR+crp 11245   [,)cico 11556   *Metcxmt 18529   fBascfbas 18532   filGencfg 18533  metUnifOLDcmetuOLD 18535  CauFiluccfilu 20914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-xmet 18538  df-fbas 18542  df-fg 18543  df-metuOLD 18544  df-fil 20472  df-ust 20828  df-cfilu 20915
This theorem is referenced by:  cfilucfil3OLD  21882  cmetcuspOLD  21918
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