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

Theorem isfbas 20157
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 4631 . . . . 5  |-  ( B  e.  A  ->  ~P B  e.  _V )
2 elpw2g 4610 . . . . 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 3122 . . . 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 18227 . . . 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 2748 . . . . . 6  |-  ( w  =  F  ->  (
w  =/=  (/)  <->  F  =/=  (/) ) )
10 neleq2 2807 . . . . . 6  |-  ( w  =  F  ->  ( (/) 
e/  w  <->  (/)  e/  F
) )
11 ineq1 3693 . . . . . . . . 9  |-  ( w  =  F  ->  (
w  i^i  ~P (
x  i^i  y )
)  =  ( F  i^i  ~P ( x  i^i  y ) ) )
1211neeq1d 2744 . . . . . . . 8  |-  ( w  =  F  ->  (
( w  i^i  ~P ( x  i^i  y
) )  =/=  (/)  <->  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )
1312raleqbi1dv 3066 . . . . . . 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 3066 . . . . . 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 1299 . . . . 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 4013 . . . . 5  |-  ( z  =  B  ->  ~P z  =  ~P B
)
1817pweqd 4015 . . . 4  |-  ( z  =  B  ->  ~P ~P z  =  ~P ~P B )
19 vex 3116 . . . . . . 7  |-  z  e. 
_V
2019pwex 4630 . . . . . 6  |-  ~P z  e.  _V
2120pwex 4630 . . . . 5  |-  ~P ~P z  e.  _V
2221a1i 11 . . . 4  |-  ( z  e.  _V  ->  ~P ~P z  e.  _V )
238, 16, 18, 22elmptrab 20155 . . 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 977 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    e/ wnel 2663   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   ` cfv 5588   fBascfbas 18217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-fbas 18227
This theorem is referenced by:  fbasne0  20158  0nelfb  20159  fbsspw  20160  isfbas2  20163  trfbas2  20171  fbasweak  20193  zfbas  20224  tsmsfbas  20453  ustfilxp  20542  minveclem3b  21670
  Copyright terms: Public domain W3C validator