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Theorem caufval 18717
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 18698 . . 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 10 . 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 4895 . . . . . 6  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 4897 . . . . 5  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 xmetf 17910 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5409 . . . . . . . 8  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
75, 6syl 15 . . . . . . 7  |-  ( D  e.  ( * Met `  X )  ->  dom  D  =  ( X  X.  X ) )
87dmeqd 4897 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
9 dmxpid 4914 . . . . . 6  |-  dom  ( X  X.  X )  =  X
108, 9syl6eq 2344 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
114, 10sylan9eqr 2350 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1211oveq1d 5889 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( dom  dom  d  ^pm  CC )  =  ( X  ^pm  CC ) )
13 simpr 447 . . . . . . . 8  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  d  =  D )
1413fveq2d 5545 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( ball `  d )  =  (
ball `  D )
)
1514oveqd 5891 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( (
f `  k )
( ball `  d )
x )  =  ( ( f `  k
) ( ball `  D
) x ) )
16 feq3 5393 . . . . . 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 15 . . . . 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 2577 . . . 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 2576 . . 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 2796 . 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 5567 . . 3  |-  ( * Met `  X ) 
C_  U. ran  * Met
2221sseli 3189 . 2  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
23 ovex 5899 . . . 4  |-  ( X 
^pm  CC )  e.  _V
2423rabex 4181 . . 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 10 . 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 5622 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801   U.cuni 3843    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   RR*cxr 8882   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   * Metcxmt 16385   ballcbl 16387   Caucca 18695
This theorem is referenced by:  iscau  18718  equivcau  18742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-xr 8887  df-xmet 16389  df-cau 18698
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