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Theorem supfil 17888
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 3338 . . . . 5  |-  ( x  =  y  ->  ( B  C_  x  <->  B  C_  y
) )
21elrab 3060 . . . 4  |-  ( y  e.  { x  e. 
~P A  |  B  C_  x }  <->  ( y  e.  ~P A  /\  B  C_  y ) )
3 vex 2927 . . . . . 6  |-  y  e. 
_V
43elpw 3773 . . . . 5  |-  ( y  e.  ~P A  <->  y  C_  A )
54anbi1i 677 . . . 4  |-  ( ( y  e.  ~P A  /\  B  C_  y )  <-> 
( y  C_  A  /\  B  C_  y ) )
62, 5bitri 241 . . 3  |-  ( y  e.  { x  e. 
~P A  |  B  C_  x }  <->  ( y  C_  A  /\  B  C_  y ) )
76a1i 11 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  (
y  e.  { x  e.  ~P A  |  B  C_  x }  <->  ( y  C_  A  /\  B  C_  y ) ) )
8 elex 2932 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
983ad2ant1 978 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  A  e.  _V )
10 simp2 958 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  B  C_  A )
11 sseq2 3338 . . . . 5  |-  ( y  =  A  ->  ( B  C_  y  <->  B  C_  A
) )
1211sbcieg 3161 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. B  C_  y  <->  B  C_  A
) )
139, 12syl 16 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  ( [. A  /  y ]. B  C_  y  <->  B  C_  A
) )
1410, 13mpbird 224 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  [. A  /  y ]. B  C_  y )
15 ss0 3626 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
1615necon3ai 2615 . . . 4  |-  ( B  =/=  (/)  ->  -.  B  C_  (/) )
17163ad2ant3 980 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  B  C_  (/) )
18 0ex 4307 . . . 4  |-  (/)  e.  _V
19 sseq2 3338 . . . 4  |-  ( y  =  (/)  ->  ( B 
C_  y  <->  B  C_  (/) ) )
2018, 19sbcie 3163 . . 3  |-  ( [. (/)  /  y ]. B  C_  y  <->  B  C_  (/) )
2117, 20sylnibr 297 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  [. (/)  /  y ]. B  C_  y )
22 sstr 3324 . . . . 5  |-  ( ( B  C_  w  /\  w  C_  z )  ->  B  C_  z )
2322expcom 425 . . . 4  |-  ( w 
C_  z  ->  ( B  C_  w  ->  B  C_  z ) )
24 vex 2927 . . . . 5  |-  w  e. 
_V
25 sseq2 3338 . . . . 5  |-  ( y  =  w  ->  ( B  C_  y  <->  B  C_  w
) )
2624, 25sbcie 3163 . . . 4  |-  ( [. w  /  y ]. B  C_  y  <->  B  C_  w )
27 vex 2927 . . . . 5  |-  z  e. 
_V
28 sseq2 3338 . . . . 5  |-  ( y  =  z  ->  ( B  C_  y  <->  B  C_  z
) )
2927, 28sbcie 3163 . . . 4  |-  ( [. z  /  y ]. B  C_  y  <->  B  C_  z )
3023, 26, 293imtr4g 262 . . 3  |-  ( w 
C_  z  ->  ( [. w  /  y ]. B  C_  y  ->  [. z  /  y ]. B  C_  y ) )
31303ad2ant3 980 . 2  |-  ( ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  /\  z  C_  A  /\  w  C_  z )  ->  ( [. w  /  y ]. B  C_  y  ->  [. z  /  y ]. B  C_  y ) )
32 ssin 3531 . . . . . 6  |-  ( ( B  C_  z  /\  B  C_  w )  <->  B  C_  (
z  i^i  w )
)
3332biimpi 187 . . . . 5  |-  ( ( B  C_  z  /\  B  C_  w )  ->  B  C_  ( z  i^i  w ) )
3429, 26, 33syl2anb 466 . . . 4  |-  ( (
[. z  /  y ]. B  C_  y  /\  [. w  /  y ]. B  C_  y )  ->  B  C_  ( z  i^i  w ) )
3527inex1 4312 . . . . 5  |-  ( z  i^i  w )  e. 
_V
36 sseq2 3338 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( B  C_  y  <->  B  C_  (
z  i^i  w )
) )
3735, 36sbcie 3163 . . . 4  |-  ( [. ( z  i^i  w
)  /  y ]. B  C_  y  <->  B  C_  (
z  i^i  w )
)
3834, 37sylibr 204 . . 3  |-  ( (
[. z  /  y ]. B  C_  y  /\  [. w  /  y ]. B  C_  y )  ->  [. ( z  i^i  w
)  /  y ]. B  C_  y )
3938a1i 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 ) )
407, 9, 14, 21, 31, 39isfild 17851 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  B  C_  x }  e.  ( Fil `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721    =/= wne 2575   {crab 2678   _Vcvv 2924   [.wsbc 3129    i^i cin 3287    C_ wss 3288   (/)c0 3596   ~Pcpw 3767   ` cfv 5421   Filcfil 17838
This theorem is referenced by:  fclscf  18018  flimfnfcls  18021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429  df-fbas 16662  df-fil 17839
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