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Theorem iscfilu 20519
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 20455 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
2 unieq 4246 . . . . . . . . 9  |-  ( u  =  U  ->  U. u  =  U. U )
32dmeqd 5196 . . . . . . . 8  |-  ( u  =  U  ->  dom  U. u  =  dom  U. U )
43fveq2d 5861 . . . . . . 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 20518 . . . . . 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 5867 . . . . . . 7  |-  ( fBas `  dom  U. U )  e.  _V
98rabex 4591 . . . . . 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 5941 . . . . 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 2530 . . 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 2896 . . . 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 20456 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
1817fveq2d 5861 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( fBas `  X )  =  (
fBas `  dom  U. U
) )
1918eleq2d 2530 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  ( fBas `  X
)  <->  F  e.  ( fBas `  dom  U. U
) ) )
2019anbi1d 704 . 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 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3469   U.cuni 4238    X. cxp 4990   dom cdm 4992   ran crn 4993   ` cfv 5579   fBascfbas 18170  UnifOncust 20430  CauFiluccfilu 20517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-ust 20431  df-cfilu 20518
This theorem is referenced by:  cfilufbas  20520  cfiluexsm  20521  fmucnd  20523  cfilufg  20524  trcfilu  20525  cfiluweak  20526  neipcfilu  20527  cfilucfilOLD  20800  cfilucfil  20801
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