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Theorem isfbas 20781
Description: The predicate " F is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
Assertion
Ref Expression
isfbas  |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_ 
~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
Distinct variable groups:    x, y, F    x, B, y
Allowed substitution hints:    A( x, y)

Proof of Theorem isfbas
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4600 . . . . 5  |-  ( B  e.  A  ->  ~P B  e.  _V )
2 elpw2g 4579 . . . . 5  |-  ( ~P B  e.  _V  ->  ( F  e.  ~P ~P B 
<->  F  C_  ~P B
) )
31, 2syl 17 . . . 4  |-  ( B  e.  A  ->  ( F  e.  ~P ~P B 
<->  F  C_  ~P B
) )
43anbi1d 709 . . 3  |-  ( B  e.  A  ->  (
( F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) )  <-> 
( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) ) ) )
5 elex 3087 . . . 4  |-  ( B  e.  A  ->  B  e.  _V )
65biantrurd 510 . . 3  |-  ( B  e.  A  ->  (
( F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) )  <-> 
( B  e.  _V  /\  ( F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) ) )
74, 6bitr3d 258 . 2  |-  ( B  e.  A  ->  (
( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )  <->  ( B  e.  _V  /\  ( F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) ) )
8 df-fbas 18908 . . . 4  |-  fBas  =  ( z  e.  _V  |->  { w  e.  ~P ~P z  |  (
w  =/=  (/)  /\  (/)  e/  w  /\  A. x  e.  w  A. y  e.  w  ( w  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) } )
9 neeq1 2703 . . . . . 6  |-  ( w  =  F  ->  (
w  =/=  (/)  <->  F  =/=  (/) ) )
10 neleq2 2763 . . . . . 6  |-  ( w  =  F  ->  ( (/) 
e/  w  <->  (/)  e/  F
) )
11 ineq1 3654 . . . . . . . . 9  |-  ( w  =  F  ->  (
w  i^i  ~P (
x  i^i  y )
)  =  ( F  i^i  ~P ( x  i^i  y ) ) )
1211neeq1d 2699 . . . . . . . 8  |-  ( w  =  F  ->  (
( w  i^i  ~P ( x  i^i  y
) )  =/=  (/)  <->  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )
1312raleqbi1dv 3031 . . . . . . 7  |-  ( w  =  F  ->  ( A. y  e.  w  ( w  i^i  ~P (
x  i^i  y )
)  =/=  (/)  <->  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )
1413raleqbi1dv 3031 . . . . . 6  |-  ( w  =  F  ->  ( A. x  e.  w  A. y  e.  w  ( w  i^i  ~P (
x  i^i  y )
)  =/=  (/)  <->  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )
159, 10, 143anbi123d 1335 . . . . 5  |-  ( w  =  F  ->  (
( w  =/=  (/)  /\  (/)  e/  w  /\  A. x  e.  w  A. y  e.  w  ( w  i^i  ~P (
x  i^i  y )
)  =/=  (/) )  <->  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) )
1615adantl 467 . . . 4  |-  ( ( z  =  B  /\  w  =  F )  ->  ( ( w  =/=  (/)  /\  (/)  e/  w  /\  A. x  e.  w  A. y  e.  w  (
w  i^i  ~P (
x  i^i  y )
)  =/=  (/) )  <->  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) )
17 pweq 3979 . . . . 5  |-  ( z  =  B  ->  ~P z  =  ~P B
)
1817pweqd 3981 . . . 4  |-  ( z  =  B  ->  ~P ~P z  =  ~P ~P B )
19 vex 3081 . . . . . . 7  |-  z  e. 
_V
2019pwex 4599 . . . . . 6  |-  ~P z  e.  _V
2120pwex 4599 . . . . 5  |-  ~P ~P z  e.  _V
2221a1i 11 . . . 4  |-  ( z  e.  _V  ->  ~P ~P z  e.  _V )
238, 16, 18, 22elmptrab 20779 . . 3  |-  ( F  e.  ( fBas `  B
)  <->  ( B  e. 
_V  /\  F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) )
24 3anass 986 . . 3  |-  ( ( B  e.  _V  /\  F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )  <->  ( B  e.  _V  /\  ( F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
2523, 24bitri 252 . 2  |-  ( F  e.  ( fBas `  B
)  <->  ( B  e. 
_V  /\  ( F  e.  ~P ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
267, 25syl6rbbr 267 1  |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_ 
~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616    e/ wnel 2617   A.wral 2773   _Vcvv 3078    i^i cin 3432    C_ wss 3433   (/)c0 3758   ~Pcpw 3976   ` cfv 5592   fBascfbas 18899
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
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-nel 2619  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-ima 4858  df-iota 5556  df-fun 5594  df-fv 5600  df-fbas 18908
This theorem is referenced by:  fbasne0  20782  0nelfb  20783  fbsspw  20784  isfbas2  20787  trfbas2  20795  fbasweak  20817  zfbas  20848  tsmsfbas  21079  ustfilxp  21164  minveclem3b  22289
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