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Theorem iscfilu 20957
Description: The predicate " F is a Cauchy filter base on uniform space  U." (Contributed by Thierry Arnoux, 18-Nov-2017.)
Assertion
Ref Expression
iscfilu  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  (CauFilu `  U )  <->  ( F  e.  ( fBas `  X
)  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
) ) )
Distinct variable groups:    v, a, F    v, U
Allowed substitution hints:    U( a)    X( v, a)

Proof of Theorem iscfilu
Dummy variables  f  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrnust 20893 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
2 unieq 4243 . . . . . . . . 9  |-  ( u  =  U  ->  U. u  =  U. U )
32dmeqd 5194 . . . . . . . 8  |-  ( u  =  U  ->  dom  U. u  =  dom  U. U )
43fveq2d 5852 . . . . . . 7  |-  ( u  =  U  ->  ( fBas `  dom  U. u
)  =  ( fBas `  dom  U. U ) )
5 raleq 3051 . . . . . . 7  |-  ( u  =  U  ->  ( A. v  e.  u  E. a  e.  f 
( a  X.  a
)  C_  v  <->  A. v  e.  U  E. a  e.  f  ( a  X.  a )  C_  v
) )
64, 5rabeqbidv 3101 . . . . . 6  |-  ( u  =  U  ->  { f  e.  ( fBas `  dom  U. u )  |  A. v  e.  u  E. a  e.  f  (
a  X.  a ) 
C_  v }  =  { f  e.  (
fBas `  dom  U. U
)  |  A. v  e.  U  E. a  e.  f  ( a  X.  a )  C_  v } )
7 df-cfilu 20956 . . . . . 6  |- CauFilu  =  ( u  e.  U. ran UnifOn  |->  { f  e.  ( fBas `  dom  U. u )  |  A. v  e.  u  E. a  e.  f  (
a  X.  a ) 
C_  v } )
8 fvex 5858 . . . . . . 7  |-  ( fBas `  dom  U. U )  e.  _V
98rabex 4588 . . . . . 6  |-  { f  e.  ( fBas `  dom  U. U )  |  A. v  e.  U  E. a  e.  f  (
a  X.  a ) 
C_  v }  e.  _V
106, 7, 9fvmpt 5931 . . . . 5  |-  ( U  e.  U. ran UnifOn  ->  (CauFilu `  U )  =  {
f  e.  ( fBas `  dom  U. U )  |  A. v  e.  U  E. a  e.  f  ( a  X.  a )  C_  v } )
111, 10syl 16 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  (CauFilu `  U
)  =  { f  e.  ( fBas `  dom  U. U )  |  A. v  e.  U  E. a  e.  f  (
a  X.  a ) 
C_  v } )
1211eleq2d 2524 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  (CauFilu `  U )  <->  F  e.  { f  e.  ( fBas `  dom  U. U )  |  A. v  e.  U  E. a  e.  f  ( a  X.  a )  C_  v } ) )
13 rexeq 3052 . . . . 5  |-  ( f  =  F  ->  ( E. a  e.  f 
( a  X.  a
)  C_  v  <->  E. a  e.  F  ( a  X.  a )  C_  v
) )
1413ralbidv 2893 . . . 4  |-  ( f  =  F  ->  ( A. v  e.  U  E. a  e.  f 
( a  X.  a
)  C_  v  <->  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
) )
1514elrab 3254 . . 3  |-  ( F  e.  { f  e.  ( fBas `  dom  U. U )  |  A. v  e.  U  E. a  e.  f  (
a  X.  a ) 
C_  v }  <->  ( F  e.  ( fBas `  dom  U. U )  /\  A. v  e.  U  E. a  e.  F  (
a  X.  a ) 
C_  v ) )
1612, 15syl6bb 261 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  (CauFilu `  U )  <->  ( F  e.  ( fBas `  dom  U. U )  /\  A. v  e.  U  E. a  e.  F  (
a  X.  a ) 
C_  v ) ) )
17 ustbas2 20894 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
1817fveq2d 5852 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( fBas `  X )  =  (
fBas `  dom  U. U
) )
1918eleq2d 2524 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  ( fBas `  X
)  <->  F  e.  ( fBas `  dom  U. U
) ) )
2019anbi1d 702 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( ( F  e.  ( fBas `  X )  /\  A. v  e.  U  E. a  e.  F  (
a  X.  a ) 
C_  v )  <->  ( F  e.  ( fBas `  dom  U. U )  /\  A. v  e.  U  E. a  e.  F  (
a  X.  a ) 
C_  v ) ) )
2116, 20bitr4d 256 1  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  (CauFilu `  U )  <->  ( F  e.  ( fBas `  X
)  /\  A. v  e.  U  E. a  e.  F  ( a  X.  a )  C_  v
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808    C_ wss 3461   U.cuni 4235    X. cxp 4986   dom cdm 4988   ran crn 4989   ` cfv 5570   fBascfbas 18601  UnifOncust 20868  CauFiluccfilu 20955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-ust 20869  df-cfilu 20956
This theorem is referenced by:  cfilufbas  20958  cfiluexsm  20959  fmucnd  20961  cfilufg  20962  trcfilu  20963  cfiluweak  20964  neipcfilu  20965  cfilucfilOLD  21238  cfilucfil  21239
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