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Theorem isfbas 19424
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 4497 . . . . 5  |-  ( B  e.  A  ->  ~P B  e.  _V )
2 elpw2g 4476 . . . . 5  |-  ( ~P B  e.  _V  ->  ( F  e.  ~P ~P B 
<->  F  C_  ~P B
) )
31, 2syl 16 . . . 4  |-  ( B  e.  A  ->  ( F  e.  ~P ~P B 
<->  F  C_  ~P B
) )
43anbi1d 704 . . 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 3002 . . . 4  |-  ( B  e.  A  ->  B  e.  _V )
65biantrurd 508 . . 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 255 . 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 17836 . . . 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 2644 . . . . . 6  |-  ( w  =  F  ->  (
w  =/=  (/)  <->  F  =/=  (/) ) )
10 neleq2 2731 . . . . . 6  |-  ( w  =  F  ->  ( (/) 
e/  w  <->  (/)  e/  F
) )
11 ineq1 3566 . . . . . . . . 9  |-  ( w  =  F  ->  (
w  i^i  ~P (
x  i^i  y )
)  =  ( F  i^i  ~P ( x  i^i  y ) ) )
1211neeq1d 2641 . . . . . . . 8  |-  ( w  =  F  ->  (
( w  i^i  ~P ( x  i^i  y
) )  =/=  (/)  <->  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )
1312raleqbi1dv 2946 . . . . . . 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 2946 . . . . . 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 1289 . . . . 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 466 . . . 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 3884 . . . . 5  |-  ( z  =  B  ->  ~P z  =  ~P B
)
1817pweqd 3886 . . . 4  |-  ( z  =  B  ->  ~P ~P z  =  ~P ~P B )
19 vex 2996 . . . . . . 7  |-  z  e. 
_V
2019pwex 4496 . . . . . 6  |-  ~P z  e.  _V
2120pwex 4496 . . . . 5  |-  ~P ~P z  e.  _V
2221a1i 11 . . . 4  |-  ( z  e.  _V  ->  ~P ~P z  e.  _V )
238, 16, 18, 22elmptrab 19422 . . 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 969 . . 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 249 . 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 264 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620    e/ wnel 2621   A.wral 2736   _Vcvv 2993    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   ` cfv 5439   fBascfbas 17826
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
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-nel 2623  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-ima 4874  df-iota 5402  df-fun 5441  df-fv 5447  df-fbas 17836
This theorem is referenced by:  fbasne0  19425  0nelfb  19426  fbsspw  19427  isfbas2  19430  trfbas2  19438  fbasweak  19460  zfbas  19491  tsmsfbas  19720  ustfilxp  19809  minveclem3b  20937
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