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Theorem supfil 19595
Description: The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
Assertion
Ref Expression
supfil  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  B  C_  x }  e.  ( Fil `  A ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem supfil
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 3481 . . . . 5  |-  ( x  =  y  ->  ( B  C_  x  <->  B  C_  y
) )
21elrab 3218 . . . 4  |-  ( y  e.  { x  e. 
~P A  |  B  C_  x }  <->  ( y  e.  ~P A  /\  B  C_  y ) )
3 selpw 3970 . . . . 5  |-  ( y  e.  ~P A  <->  y  C_  A )
43anbi1i 695 . . . 4  |-  ( ( y  e.  ~P A  /\  B  C_  y )  <-> 
( y  C_  A  /\  B  C_  y ) )
52, 4bitri 249 . . 3  |-  ( y  e.  { x  e. 
~P A  |  B  C_  x }  <->  ( y  C_  A  /\  B  C_  y ) )
65a1i 11 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  (
y  e.  { x  e.  ~P A  |  B  C_  x }  <->  ( y  C_  A  /\  B  C_  y ) ) )
7 elex 3081 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
873ad2ant1 1009 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  A  e.  _V )
9 simp2 989 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  B  C_  A )
10 sseq2 3481 . . . . 5  |-  ( y  =  A  ->  ( B  C_  y  <->  B  C_  A
) )
1110sbcieg 3321 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. B  C_  y  <->  B  C_  A
) )
128, 11syl 16 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  ( [. A  /  y ]. B  C_  y  <->  B  C_  A
) )
139, 12mpbird 232 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  [. A  /  y ]. B  C_  y )
14 ss0 3771 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
1514necon3ai 2677 . . . 4  |-  ( B  =/=  (/)  ->  -.  B  C_  (/) )
16153ad2ant3 1011 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  B  C_  (/) )
17 0ex 4525 . . . 4  |-  (/)  e.  _V
18 sseq2 3481 . . . 4  |-  ( y  =  (/)  ->  ( B 
C_  y  <->  B  C_  (/) ) )
1917, 18sbcie 3323 . . 3  |-  ( [. (/)  /  y ]. B  C_  y  <->  B  C_  (/) )
2016, 19sylnibr 305 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  [. (/)  /  y ]. B  C_  y )
21 sstr 3467 . . . . 5  |-  ( ( B  C_  w  /\  w  C_  z )  ->  B  C_  z )
2221expcom 435 . . . 4  |-  ( w 
C_  z  ->  ( B  C_  w  ->  B  C_  z ) )
23 vex 3075 . . . . 5  |-  w  e. 
_V
24 sseq2 3481 . . . . 5  |-  ( y  =  w  ->  ( B  C_  y  <->  B  C_  w
) )
2523, 24sbcie 3323 . . . 4  |-  ( [. w  /  y ]. B  C_  y  <->  B  C_  w )
26 vex 3075 . . . . 5  |-  z  e. 
_V
27 sseq2 3481 . . . . 5  |-  ( y  =  z  ->  ( B  C_  y  <->  B  C_  z
) )
2826, 27sbcie 3323 . . . 4  |-  ( [. z  /  y ]. B  C_  y  <->  B  C_  z )
2922, 25, 283imtr4g 270 . . 3  |-  ( w 
C_  z  ->  ( [. w  /  y ]. B  C_  y  ->  [. z  /  y ]. B  C_  y ) )
30293ad2ant3 1011 . 2  |-  ( ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  /\  z  C_  A  /\  w  C_  z )  ->  ( [. w  /  y ]. B  C_  y  ->  [. z  /  y ]. B  C_  y ) )
31 ssin 3675 . . . . . 6  |-  ( ( B  C_  z  /\  B  C_  w )  <->  B  C_  (
z  i^i  w )
)
3231biimpi 194 . . . . 5  |-  ( ( B  C_  z  /\  B  C_  w )  ->  B  C_  ( z  i^i  w ) )
3328, 25, 32syl2anb 479 . . . 4  |-  ( (
[. z  /  y ]. B  C_  y  /\  [. w  /  y ]. B  C_  y )  ->  B  C_  ( z  i^i  w ) )
3426inex1 4536 . . . . 5  |-  ( z  i^i  w )  e. 
_V
35 sseq2 3481 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( B  C_  y  <->  B  C_  (
z  i^i  w )
) )
3634, 35sbcie 3323 . . . 4  |-  ( [. ( z  i^i  w
)  /  y ]. B  C_  y  <->  B  C_  (
z  i^i  w )
)
3733, 36sylibr 212 . . 3  |-  ( (
[. z  /  y ]. B  C_  y  /\  [. w  /  y ]. B  C_  y )  ->  [. ( z  i^i  w
)  /  y ]. B  C_  y )
3837a1i 11 . 2  |-  ( ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  /\  z  C_  A  /\  w  C_  A )  ->  (
( [. z  /  y ]. B  C_  y  /\  [. w  /  y ]. B  C_  y )  ->  [. ( z  i^i  w
)  /  y ]. B  C_  y ) )
396, 8, 13, 20, 30, 38isfild 19558 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  B  C_  x }  e.  ( Fil `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2645   {crab 2800   _Vcvv 3072   [.wsbc 3288    i^i cin 3430    C_ wss 3431   (/)c0 3740   ~Pcpw 3963   ` cfv 5521   Filcfil 19545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-fbas 17934  df-fil 19546
This theorem is referenced by:  fclscf  19725  flimfnfcls  19728
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