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Theorem caufval 19101
Description: The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)
Assertion
Ref Expression
caufval  |-  ( D  e.  ( * Met `  X )  ->  ( Cau `  D )  =  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  (
f  |`  ( ZZ>= `  k
) ) : (
ZZ>= `  k ) --> ( ( f `  k
) ( ball `  D
) x ) } )
Distinct variable groups:    f, k, x, D    f, X, k, x

Proof of Theorem caufval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 df-cau 19082 . . 3  |-  Cau  =  ( d  e.  U. ran  * Met  |->  { f  e.  ( dom  dom  d  ^pm  CC )  | 
A. x  e.  RR+  E. k  e.  ZZ  (
f  |`  ( ZZ>= `  k
) ) : (
ZZ>= `  k ) --> ( ( f `  k
) ( ball `  d
) x ) } )
21a1i 11 . 2  |-  ( D  e.  ( * Met `  X )  ->  Cau  =  ( d  e. 
U. ran  * Met  |->  { f  e.  ( dom  dom  d  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  d
) x ) } ) )
3 dmeq 5012 . . . . . 6  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5014 . . . . 5  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 xmetf 18270 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5537 . . . . . . . 8  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
75, 6syl 16 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  dom  D  =  ( X  X.  X ) )
87dmeqd 5014 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
9 dmxpid 5031 . . . . . 6  |-  dom  ( X  X.  X )  =  X
108, 9syl6eq 2437 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
114, 10sylan9eqr 2443 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1211oveq1d 6037 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( dom  dom  d  ^pm  CC )  =  ( X  ^pm  CC ) )
13 simpr 448 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  d  =  D )
1413fveq2d 5674 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( ball `  d )  =  (
ball `  D )
)
1514oveqd 6039 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( (
f `  k )
( ball `  d )
x )  =  ( ( f `  k
) ( ball `  D
) x ) )
16 feq3 5520 . . . . . 6  |-  ( ( ( f `  k
) ( ball `  d
) x )  =  ( ( f `  k ) ( ball `  D ) x )  ->  ( ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  d
) x )  <->  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) ) )
1715, 16syl 16 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( (
f  |`  ( ZZ>= `  k
) ) : (
ZZ>= `  k ) --> ( ( f `  k
) ( ball `  d
) x )  <->  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) ) )
1817rexbidv 2672 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( E. k  e.  ZZ  (
f  |`  ( ZZ>= `  k
) ) : (
ZZ>= `  k ) --> ( ( f `  k
) ( ball `  d
) x )  <->  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) ) )
1918ralbidv 2671 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  d
) x )  <->  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) ) )
2012, 19rabeqbidv 2896 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  { f  e.  ( dom  dom  d  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  d
) x ) }  =  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) } )
21 fvssunirn 5696 . . 3  |-  ( * Met `  X ) 
C_  U. ran  * Met
2221sseli 3289 . 2  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
23 ovex 6047 . . . 4  |-  ( X 
^pm  CC )  e.  _V
2423rabex 4297 . . 3  |-  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) }  e.  _V
2524a1i 11 . 2  |-  ( D  e.  ( * Met `  X )  ->  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  ( f  |`  ( ZZ>= `  k )
) : ( ZZ>= `  k ) --> ( ( f `  k ) ( ball `  D
) x ) }  e.  _V )
262, 20, 22, 25fvmptd 5751 1  |-  ( D  e.  ( * Met `  X )  ->  ( Cau `  D )  =  { f  e.  ( X  ^pm  CC )  |  A. x  e.  RR+  E. k  e.  ZZ  (
f  |`  ( ZZ>= `  k
) ) : (
ZZ>= `  k ) --> ( ( f `  k
) ( ball `  D
) x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652   {crab 2655   _Vcvv 2901   U.cuni 3959    e. cmpt 4209    X. cxp 4818   dom cdm 4820   ran crn 4821    |` cres 4822   -->wf 5392   ` cfv 5396  (class class class)co 6022    ^pm cpm 6957   CCcc 8923   RR*cxr 9054   ZZcz 10216   ZZ>=cuz 10422   RR+crp 10546   * Metcxmt 16614   ballcbl 16616   Caucca 19079
This theorem is referenced by:  iscau  19102  equivcau  19126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-map 6958  df-xr 9059  df-xmet 16621  df-cau 19082
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