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Theorem caufval 21840
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 21821 . . 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 5213 . . . . . 6  |-  ( d  =  D  ->  dom  d  =  dom  D )
43dmeqd 5215 . . . . 5  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
5 xmetf 20958 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
6 fdm 5741 . . . . . . . 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 5215 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
9 dmxpid 5232 . . . . . 6  |-  dom  ( X  X.  X )  =  X
108, 9syl6eq 2514 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  X )
114, 10sylan9eqr 2520 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1211oveq1d 6311 . . 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 5876 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( ball `  d )  =  (
ball `  D )
)
1514oveqd 6313 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( (
f `  k )
( ball `  d )
x )  =  ( ( f `  k
) ( ball `  D
) x ) )
1615feq3d 5725 . . . . 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 2968 . . . 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 2896 . . 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 3104 . 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 5895 . . 3  |-  ( *Met `  X ) 
C_  U. ran  *Met
2120sseli 3495 . 2  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
22 ovex 6324 . . . 4  |-  ( X 
^pm  CC )  e.  _V
2322rabex 4607 . . 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 5961 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 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109   U.cuni 4251    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^pm cpm 7439   CCcc 9507   RR*cxr 9644   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   *Metcxmt 18530   ballcbl 18532   Caucca 21818
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2812  df-rex 2813  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-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-xr 9649  df-xmet 18539  df-cau 21821
This theorem is referenced by:  iscau  21841  equivcau  21865
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