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Theorem isfbas2 17546
Description: The predicate " F is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
isfbas2  |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_ 
~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y ) ) ) ) )
Distinct variable groups:    x, y,
z, F    x, B, y, z
Allowed substitution hints:    A( x, y, z)

Proof of Theorem isfbas2
StepHypRef Expression
1 isfbas 17540 . 2  |-  ( 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 )
)  =/=  (/) ) ) ) )
2 elin 3371 . . . . . . . 8  |-  ( z  e.  ( F  i^i  ~P ( x  i^i  y
) )  <->  ( z  e.  F  /\  z  e.  ~P ( x  i^i  y ) ) )
3 vex 2804 . . . . . . . . . 10  |-  z  e. 
_V
43elpw 3644 . . . . . . . . 9  |-  ( z  e.  ~P ( x  i^i  y )  <->  z  C_  ( x  i^i  y
) )
54anbi2i 675 . . . . . . . 8  |-  ( ( z  e.  F  /\  z  e.  ~P (
x  i^i  y )
)  <->  ( z  e.  F  /\  z  C_  ( x  i^i  y
) ) )
62, 5bitri 240 . . . . . . 7  |-  ( z  e.  ( F  i^i  ~P ( x  i^i  y
) )  <->  ( z  e.  F  /\  z  C_  ( x  i^i  y
) ) )
76exbii 1572 . . . . . 6  |-  ( E. z  z  e.  ( F  i^i  ~P (
x  i^i  y )
)  <->  E. z ( z  e.  F  /\  z  C_  ( x  i^i  y
) ) )
8 n0 3477 . . . . . 6  |-  ( ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/)  <->  E. z 
z  e.  ( F  i^i  ~P ( x  i^i  y ) ) )
9 df-rex 2562 . . . . . 6  |-  ( E. z  e.  F  z 
C_  ( x  i^i  y )  <->  E. z
( z  e.  F  /\  z  C_  ( x  i^i  y ) ) )
107, 8, 93bitr4i 268 . . . . 5  |-  ( ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/)  <->  E. z  e.  F  z  C_  ( x  i^i  y
) )
11102ralbii 2582 . . . 4  |-  ( A. x  e.  F  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/)  <->  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y ) )
12113anbi3i 1144 . . 3  |-  ( ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) )  <->  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y ) ) )
1312anbi2i 675 . 2  |-  ( ( F  C_  ~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  E. z  e.  F  z  C_  ( x  i^i  y
) ) ) )
141, 13syl6bb 252 1  |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_ 
~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    e. wcel 1696    =/= wne 2459    e/ wnel 2460   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   ` cfv 5271   fBascfbas 17534
This theorem is referenced by:  fbasssin  17547  fbun  17551  opnfbas  17553  isfil2  17567  fsubbas  17578  fbasrn  17595  rnelfmlem  17663  tailfb  26429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-fbas 17536
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