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Theorem isfbas 17524
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 4194 . . . . 5  |-  ( B  e.  A  ->  ~P B  e.  _V )
2 elpw2g 4174 . . . . 5  |-  ( ~P B  e.  _V  ->  ( F  e.  ~P ~P B 
<->  F  C_  ~P B
) )
31, 2syl 15 . . . 4  |-  ( B  e.  A  ->  ( F  e.  ~P ~P B 
<->  F  C_  ~P B
) )
43anbi1d 685 . . 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 2796 . . . 4  |-  ( B  e.  A  ->  B  e.  _V )
65biantrurd 494 . . 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 246 . 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 17520 . . . 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 2454 . . . . . 6  |-  ( w  =  F  ->  (
w  =/=  (/)  <->  F  =/=  (/) ) )
10 neleq2 2538 . . . . . 6  |-  ( w  =  F  ->  ( (/) 
e/  w  <->  (/)  e/  F
) )
11 ineq1 3363 . . . . . . . . 9  |-  ( w  =  F  ->  (
w  i^i  ~P (
x  i^i  y )
)  =  ( F  i^i  ~P ( x  i^i  y ) ) )
1211neeq1d 2459 . . . . . . . 8  |-  ( w  =  F  ->  (
( w  i^i  ~P ( x  i^i  y
) )  =/=  (/)  <->  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) )
1312raleqbi1dv 2744 . . . . . . 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 2744 . . . . . 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 1252 . . . . 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 452 . . . 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 3628 . . . . 5  |-  ( z  =  B  ->  ~P z  =  ~P B
)
1817pweqd 3630 . . . 4  |-  ( z  =  B  ->  ~P ~P z  =  ~P ~P B )
19 vex 2791 . . . . . . 7  |-  z  e. 
_V
2019pwex 4193 . . . . . 6  |-  ~P z  e.  _V
2120pwex 4193 . . . . 5  |-  ~P ~P z  e.  _V
2221a1i 10 . . . 4  |-  ( z  e.  _V  ->  ~P ~P z  e.  _V )
238, 16, 18, 22elmptrab 17522 . . 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 938 . . 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 240 . 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 255 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   ` cfv 5255   fBascfbas 17518
This theorem is referenced by:  fbasne0  17525  0nelfb  17526  fbsspw  17527  isfbas2  17530  trfbas2  17538  fbasweak  17560  zfbas  17591  tsmsfbas  17810  minveclem3b  18792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fbas 17520
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