MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscfilu Structured version   Unicode version

Theorem iscfilu 21227
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 21163 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  U  e.  U.
ran UnifOn )
2 unieq 4221 . . . . . . . . 9  |-  ( u  =  U  ->  U. u  =  U. U )
32dmeqd 5048 . . . . . . . 8  |-  ( u  =  U  ->  dom  U. u  =  dom  U. U )
43fveq2d 5876 . . . . . . 7  |-  ( u  =  U  ->  ( fBas `  dom  U. u
)  =  ( fBas `  dom  U. U ) )
5 raleq 3023 . . . . . . 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 3073 . . . . . 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 21226 . . . . . 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 5882 . . . . . . 7  |-  ( fBas `  dom  U. U )  e.  _V
98rabex 4567 . . . . . 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 5955 . . . . 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 17 . . . 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 2490 . . 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 3024 . . . . 5  |-  ( f  =  F  ->  ( E. a  e.  f 
( a  X.  a
)  C_  v  <->  E. a  e.  F  ( a  X.  a )  C_  v
) )
1413ralbidv 2862 . . . 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 3226 . . 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 264 . 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 21164 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  X  =  dom  U. U )
1817fveq2d 5876 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( fBas `  X )  =  (
fBas `  dom  U. U
) )
1918eleq2d 2490 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( F  e.  ( fBas `  X
)  <->  F  e.  ( fBas `  dom  U. U
) ) )
2019anbi1d 709 . 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 259 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 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   {crab 2777    C_ wss 3433   U.cuni 4213    X. cxp 4843   dom cdm 4845   ran crn 4846   ` cfv 5592   fBascfbas 18886  UnifOncust 21138  CauFiluccfilu 21225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600  df-ust 21139  df-cfilu 21226
This theorem is referenced by:  cfilufbas  21228  cfiluexsm  21229  fmucnd  21231  cfilufg  21232  trcfilu  21233  cfiluweak  21234  neipcfilu  21235  cfilucfil  21498
  Copyright terms: Public domain W3C validator