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Theorem supfil 20902
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 3487 . . . . 5  |-  ( x  =  y  ->  ( B  C_  x  <->  B  C_  y
) )
21elrab 3230 . . . 4  |-  ( y  e.  { x  e. 
~P A  |  B  C_  x }  <->  ( y  e.  ~P A  /\  B  C_  y ) )
3 selpw 3987 . . . . 5  |-  ( y  e.  ~P A  <->  y  C_  A )
43anbi1i 700 . . . 4  |-  ( ( y  e.  ~P A  /\  B  C_  y )  <-> 
( y  C_  A  /\  B  C_  y ) )
52, 4bitri 253 . . 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 3091 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
873ad2ant1 1027 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  A  e.  _V )
9 simp2 1007 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  B  C_  A )
10 sseq2 3487 . . . . 5  |-  ( y  =  A  ->  ( B  C_  y  <->  B  C_  A
) )
1110sbcieg 3333 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  y ]. B  C_  y  <->  B  C_  A
) )
128, 11syl 17 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  ( [. A  /  y ]. B  C_  y  <->  B  C_  A
) )
139, 12mpbird 236 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  [. A  /  y ]. B  C_  y )
14 ss0 3794 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
1514necon3ai 2653 . . . 4  |-  ( B  =/=  (/)  ->  -.  B  C_  (/) )
16153ad2ant3 1029 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  B  C_  (/) )
17 0ex 4554 . . . 4  |-  (/)  e.  _V
18 sseq2 3487 . . . 4  |-  ( y  =  (/)  ->  ( B 
C_  y  <->  B  C_  (/) ) )
1917, 18sbcie 3335 . . 3  |-  ( [. (/)  /  y ]. B  C_  y  <->  B  C_  (/) )
2016, 19sylnibr 307 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  [. (/)  /  y ]. B  C_  y )
21 sstr 3473 . . . . 5  |-  ( ( B  C_  w  /\  w  C_  z )  ->  B  C_  z )
2221expcom 437 . . . 4  |-  ( w 
C_  z  ->  ( B  C_  w  ->  B  C_  z ) )
23 vex 3085 . . . . 5  |-  w  e. 
_V
24 sseq2 3487 . . . . 5  |-  ( y  =  w  ->  ( B  C_  y  <->  B  C_  w
) )
2523, 24sbcie 3335 . . . 4  |-  ( [. w  /  y ]. B  C_  y  <->  B  C_  w )
26 vex 3085 . . . . 5  |-  z  e. 
_V
27 sseq2 3487 . . . . 5  |-  ( y  =  z  ->  ( B  C_  y  <->  B  C_  z
) )
2826, 27sbcie 3335 . . . 4  |-  ( [. z  /  y ]. B  C_  y  <->  B  C_  z )
2922, 25, 283imtr4g 274 . . 3  |-  ( w 
C_  z  ->  ( [. w  /  y ]. B  C_  y  ->  [. z  /  y ]. B  C_  y ) )
30293ad2ant3 1029 . 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 3685 . . . . . 6  |-  ( ( B  C_  z  /\  B  C_  w )  <->  B  C_  (
z  i^i  w )
)
3231biimpi 198 . . . . 5  |-  ( ( B  C_  z  /\  B  C_  w )  ->  B  C_  ( z  i^i  w ) )
3328, 25, 32syl2anb 482 . . . 4  |-  ( (
[. z  /  y ]. B  C_  y  /\  [. w  /  y ]. B  C_  y )  ->  B  C_  ( z  i^i  w ) )
3426inex1 4563 . . . . 5  |-  ( z  i^i  w )  e. 
_V
35 sseq2 3487 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( B  C_  y  <->  B  C_  (
z  i^i  w )
) )
3634, 35sbcie 3335 . . . 4  |-  ( [. ( z  i^i  w
)  /  y ]. B  C_  y  <->  B  C_  (
z  i^i  w )
)
3733, 36sylibr 216 . . 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 20865 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 188    /\ wa 371    /\ w3a 983    e. wcel 1869    =/= wne 2619   {crab 2780   _Vcvv 3082   [.wsbc 3300    i^i cin 3436    C_ wss 3437   (/)c0 3762   ~Pcpw 3980   ` cfv 5599   Filcfil 20852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fv 5607  df-fbas 18960  df-fil 20853
This theorem is referenced by:  fclscf  21032  flimfnfcls  21035
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