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Theorem iscfilu 19885
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 19821 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
2 unieq 4120 . . . . . . . . 9  |-  ( u  =  U  ->  U. u  =  U. U )
32dmeqd 5063 . . . . . . . 8  |-  ( u  =  U  ->  dom  U. u  =  dom  U. U )
43fveq2d 5716 . . . . . . 7  |-  ( u  =  U  ->  ( fBas `  dom  U. u
)  =  ( fBas `  dom  U. U ) )
5 raleq 2938 . . . . . . 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 2988 . . . . . 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 19884 . . . . . 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 5722 . . . . . . 7  |-  ( fBas `  dom  U. U )  e.  _V
98rabex 4464 . . . . . 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 5795 . . . . 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 2510 . . 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 2939 . . . . 5  |-  ( f  =  F  ->  ( E. a  e.  f 
( a  X.  a
)  C_  v  <->  E. a  e.  F  ( a  X.  a )  C_  v
) )
1413ralbidv 2756 . . . 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 3138 . . 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 19822 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
1817fveq2d 5716 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( fBas `  X )  =  (
fBas `  dom  U. U
) )
1918eleq2d 2510 . . 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 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740    C_ wss 3349   U.cuni 4112    X. cxp 4859   dom cdm 4861   ran crn 4862   ` cfv 5439   fBascfbas 17826  UnifOncust 19796  CauFiluccfilu 19883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-iota 5402  df-fun 5441  df-fn 5442  df-fv 5447  df-ust 19797  df-cfilu 19884
This theorem is referenced by:  cfilufbas  19886  cfiluexsm  19887  fmucnd  19889  cfilufg  19890  trcfilu  19891  cfiluweak  19892  neipcfilu  19893  cfilucfilOLD  20166  cfilucfil  20167
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