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Theorem caufval 22323
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 22304 . . 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 5040 . . . . . 6  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5042 . . . . 5  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 xmetf 21422 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5745 . . . . . . . 8  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
75, 6syl 17 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  dom  D  =  ( X  X.  X ) )
87dmeqd 5042 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
9 dmxpid 5060 . . . . . 6  |-  dom  ( X  X.  X )  =  X
108, 9syl6eq 2521 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  X )
114, 10sylan9eqr 2527 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1211oveq1d 6323 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( dom  dom  d  ^pm  CC )  =  ( X  ^pm  CC ) )
13 simpr 468 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  d  =  D )
1413fveq2d 5883 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( ball `  d )  =  (
ball `  D )
)
1514oveqd 6325 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( (
f `  k )
( ball `  d )
x )  =  ( ( f `  k
) ( ball `  D
) x ) )
1615feq3d 5726 . . . . 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 ) ) )
1716rexbidv 2892 . . . 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 ) ) )
1817ralbidv 2829 . . 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 ) ) )
1912, 18rabeqbidv 3026 . 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 ) } )
20 fvssunirn 5902 . . 3  |-  ( *Met `  X ) 
C_  U. ran  *Met
2120sseli 3414 . 2  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
22 ovex 6336 . . . 4  |-  ( X 
^pm  CC )  e.  _V
2322rabex 4550 . . 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
2423a1i 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 )
252, 19, 21, 24fvmptd 5969 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    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   U.cuni 4190    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^pm cpm 7491   CCcc 9555   RR*cxr 9692   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   *Metcxmt 19032   ballcbl 19034   Caucca 22301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-xr 9697  df-xmet 19040  df-cau 22304
This theorem is referenced by:  iscau  22324  equivcau  22348
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