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Theorem isfbas2 20087
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 20081 . 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 3687 . . . . . . . 8  |-  ( z  e.  ( F  i^i  ~P ( x  i^i  y
) )  <->  ( z  e.  F  /\  z  e.  ~P ( x  i^i  y ) ) )
3 selpw 4017 . . . . . . . . 9  |-  ( z  e.  ~P ( x  i^i  y )  <->  z  C_  ( x  i^i  y
) )
43anbi2i 694 . . . . . . . 8  |-  ( ( z  e.  F  /\  z  e.  ~P (
x  i^i  y )
)  <->  ( z  e.  F  /\  z  C_  ( x  i^i  y
) ) )
52, 4bitri 249 . . . . . . 7  |-  ( z  e.  ( F  i^i  ~P ( x  i^i  y
) )  <->  ( z  e.  F  /\  z  C_  ( x  i^i  y
) ) )
65exbii 1644 . . . . . 6  |-  ( E. z  z  e.  ( F  i^i  ~P (
x  i^i  y )
)  <->  E. z ( z  e.  F  /\  z  C_  ( x  i^i  y
) ) )
7 n0 3794 . . . . . 6  |-  ( ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/)  <->  E. z 
z  e.  ( F  i^i  ~P ( x  i^i  y ) ) )
8 df-rex 2820 . . . . . 6  |-  ( E. z  e.  F  z 
C_  ( x  i^i  y )  <->  E. z
( z  e.  F  /\  z  C_  ( x  i^i  y ) ) )
96, 7, 83bitr4i 277 . . . . 5  |-  ( ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/)  <->  E. z  e.  F  z  C_  ( x  i^i  y
) )
1092ralbii 2896 . . . 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 ) )
11103anbi3i 1189 . . 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 ) ) )
1211anbi2i 694 . 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
) ) ) )
131, 12syl6bb 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   E.wex 1596    e. wcel 1767    =/= wne 2662    e/ wnel 2663   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   ` cfv 5587   fBascfbas 18193
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 5550  df-fun 5589  df-fv 5595  df-fbas 18203
This theorem is referenced by:  fbasssin  20088  fbun  20092  opnfbas  20094  isfil2  20108  fsubbas  20119  fbasrn  20136  rnelfmlem  20204  metustfbasOLD  20819  metustfbas  20820  tailfb  29814
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