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Theorem caufval 20902
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 20883 . . 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 5138 . . . . . 6  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5140 . . . . 5  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 xmetf 20020 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5661 . . . . . . . 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 5140 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
9 dmxpid 5157 . . . . . 6  |-  dom  ( X  X.  X )  =  X
108, 9syl6eq 2508 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  X )
114, 10sylan9eqr 2514 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1211oveq1d 6205 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( dom  dom  d  ^pm  CC )  =  ( X  ^pm  CC ) )
13 simpr 461 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  d  =  D )
1413fveq2d 5793 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( ball `  d )  =  (
ball `  D )
)
1514oveqd 6207 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( (
f `  k )
( ball `  d )
x )  =  ( ( f `  k
) ( ball `  D
) x ) )
16 feq3 5642 . . . . . 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 2844 . . . 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 2839 . . 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 3063 . 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 5812 . . 3  |-  ( *Met `  X ) 
C_  U. ran  *Met
2221sseli 3450 . 2  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
23 ovex 6215 . . . 4  |-  ( X 
^pm  CC )  e.  _V
2423rabex 4541 . . 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 5878 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3068   U.cuni 4189    |-> cmpt 4448    X. cxp 4936   dom cdm 4938   ran crn 4939    |` cres 4940   -->wf 5512   ` cfv 5516  (class class class)co 6190    ^pm cpm 7315   CCcc 9381   RR*cxr 9518   ZZcz 10747   ZZ>=cuz 10962   RR+crp 11092   *Metcxmt 17910   ballcbl 17912   Caucca 20880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-map 7316  df-xr 9523  df-xmet 17919  df-cau 20883
This theorem is referenced by:  iscau  20903  equivcau  20927
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