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Theorem supfil 20522
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 3521 . . . . 5  |-  ( x  =  y  ->  ( B  C_  x  <->  B  C_  y
) )
21elrab 3257 . . . 4  |-  ( y  e.  { x  e. 
~P A  |  B  C_  x }  <->  ( y  e.  ~P A  /\  B  C_  y ) )
3 selpw 4022 . . . . 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 3118 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
873ad2ant1 1017 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  A  e.  _V )
9 simp2 997 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  B  C_  A )
10 sseq2 3521 . . . . 5  |-  ( y  =  A  ->  ( B  C_  y  <->  B  C_  A
) )
1110sbcieg 3360 . . . 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 3825 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
1514necon3ai 2685 . . . 4  |-  ( B  =/=  (/)  ->  -.  B  C_  (/) )
16153ad2ant3 1019 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  B  C_  (/) )
17 0ex 4587 . . . 4  |-  (/)  e.  _V
18 sseq2 3521 . . . 4  |-  ( y  =  (/)  ->  ( B 
C_  y  <->  B  C_  (/) ) )
1917, 18sbcie 3362 . . 3  |-  ( [. (/)  /  y ]. B  C_  y  <->  B  C_  (/) )
2016, 19sylnibr 305 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  [. (/)  /  y ]. B  C_  y )
21 sstr 3507 . . . . 5  |-  ( ( B  C_  w  /\  w  C_  z )  ->  B  C_  z )
2221expcom 435 . . . 4  |-  ( w 
C_  z  ->  ( B  C_  w  ->  B  C_  z ) )
23 vex 3112 . . . . 5  |-  w  e. 
_V
24 sseq2 3521 . . . . 5  |-  ( y  =  w  ->  ( B  C_  y  <->  B  C_  w
) )
2523, 24sbcie 3362 . . . 4  |-  ( [. w  /  y ]. B  C_  y  <->  B  C_  w )
26 vex 3112 . . . . 5  |-  z  e. 
_V
27 sseq2 3521 . . . . 5  |-  ( y  =  z  ->  ( B  C_  y  <->  B  C_  z
) )
2826, 27sbcie 3362 . . . 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 1019 . 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 3716 . . . . . 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 4597 . . . . 5  |-  ( z  i^i  w )  e. 
_V
35 sseq2 3521 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( B  C_  y  <->  B  C_  (
z  i^i  w )
) )
3634, 35sbcie 3362 . . . 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 20485 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 973    e. wcel 1819    =/= wne 2652   {crab 2811   _Vcvv 3109   [.wsbc 3327    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   ` cfv 5594   Filcfil 20472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-fbas 18543  df-fil 20473
This theorem is referenced by:  fclscf  20652  flimfnfcls  20655
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