MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supfil Structured version   Unicode version

Theorem supfil 17965
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 3359 . . . . 5  |-  ( x  =  y  ->  ( B  C_  x  <->  B  C_  y
) )
21elrab 3101 . . . 4  |-  ( y  e.  { x  e. 
~P A  |  B  C_  x }  <->  ( y  e.  ~P A  /\  B  C_  y ) )
3 vex 2968 . . . . . 6  |-  y  e. 
_V
43elpw 3834 . . . . 5  |-  ( y  e.  ~P A  <->  y  C_  A )
54anbi1i 678 . . . 4  |-  ( ( y  e.  ~P A  /\  B  C_  y )  <-> 
( y  C_  A  /\  B  C_  y ) )
62, 5bitri 242 . . 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 2973 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
983ad2ant1 979 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  A  e.  _V )
10 simp2 959 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  B  C_  A )
11 sseq2 3359 . . . . 5  |-  ( y  =  A  ->  ( B  C_  y  <->  B  C_  A
) )
1211sbcieg 3202 . . . 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 225 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  [. A  /  y ]. B  C_  y )
15 ss0 3646 . . . . 5  |-  ( B 
C_  (/)  ->  B  =  (/) )
1615necon3ai 2651 . . . 4  |-  ( B  =/=  (/)  ->  -.  B  C_  (/) )
17163ad2ant3 981 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  B  C_  (/) )
18 0ex 4370 . . . 4  |-  (/)  e.  _V
19 sseq2 3359 . . . 4  |-  ( y  =  (/)  ->  ( B 
C_  y  <->  B  C_  (/) ) )
2018, 19sbcie 3204 . . 3  |-  ( [. (/)  /  y ]. B  C_  y  <->  B  C_  (/) )
2117, 20sylnibr 298 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  -.  [. (/)  /  y ]. B  C_  y )
22 sstr 3345 . . . . 5  |-  ( ( B  C_  w  /\  w  C_  z )  ->  B  C_  z )
2322expcom 426 . . . 4  |-  ( w 
C_  z  ->  ( B  C_  w  ->  B  C_  z ) )
24 vex 2968 . . . . 5  |-  w  e. 
_V
25 sseq2 3359 . . . . 5  |-  ( y  =  w  ->  ( B  C_  y  <->  B  C_  w
) )
2624, 25sbcie 3204 . . . 4  |-  ( [. w  /  y ]. B  C_  y  <->  B  C_  w )
27 vex 2968 . . . . 5  |-  z  e. 
_V
28 sseq2 3359 . . . . 5  |-  ( y  =  z  ->  ( B  C_  y  <->  B  C_  z
) )
2927, 28sbcie 3204 . . . 4  |-  ( [. z  /  y ]. B  C_  y  <->  B  C_  z )
3023, 26, 293imtr4g 263 . . 3  |-  ( w 
C_  z  ->  ( [. w  /  y ]. B  C_  y  ->  [. z  /  y ]. B  C_  y ) )
31303ad2ant3 981 . 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 3551 . . . . . 6  |-  ( ( B  C_  z  /\  B  C_  w )  <->  B  C_  (
z  i^i  w )
)
3332biimpi 188 . . . . 5  |-  ( ( B  C_  z  /\  B  C_  w )  ->  B  C_  ( z  i^i  w ) )
3429, 26, 33syl2anb 467 . . . 4  |-  ( (
[. z  /  y ]. B  C_  y  /\  [. w  /  y ]. B  C_  y )  ->  B  C_  ( z  i^i  w ) )
3527inex1 4379 . . . . 5  |-  ( z  i^i  w )  e. 
_V
36 sseq2 3359 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( B  C_  y  <->  B  C_  (
z  i^i  w )
) )
3735, 36sbcie 3204 . . . 4  |-  ( [. ( z  i^i  w
)  /  y ]. B  C_  y  <->  B  C_  (
z  i^i  w )
)
3834, 37sylibr 205 . . 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 17928 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 178    /\ wa 360    /\ w3a 937    e. wcel 1728    =/= wne 2606   {crab 2716   _Vcvv 2965   [.wsbc 3170    i^i cin 3308    C_ wss 3309   (/)c0 3616   ~Pcpw 3828   ` cfv 5489   Filcfil 17915
This theorem is referenced by:  fclscf  18095  flimfnfcls  18098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fv 5497  df-fbas 16737  df-fil 17916
  Copyright terms: Public domain W3C validator