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

Theorem csdfil 20130
Description: The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
csdfil  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  { x  e.  ~P X  |  ( X  \  x )  ~<  X }  e.  ( Fil `  X
) )
Distinct variable group:    x, X

Proof of Theorem csdfil
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 3616 . . . . . 6  |-  ( x  =  y  ->  ( X  \  x )  =  ( X  \  y
) )
21breq1d 4457 . . . . 5  |-  ( x  =  y  ->  (
( X  \  x
)  ~<  X  <->  ( X  \  y )  ~<  X ) )
32elrab 3261 . . . 4  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  e.  ~P X  /\  ( X  \  y )  ~<  X ) )
4 selpw 4017 . . . . 5  |-  ( y  e.  ~P X  <->  y  C_  X )
54anbi1i 695 . . . 4  |-  ( ( y  e.  ~P X  /\  ( X  \  y
)  ~<  X )  <->  ( y  C_  X  /\  ( X 
\  y )  ~<  X ) )
63, 5bitri 249 . . 3  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  C_  X  /\  ( X 
\  y )  ~<  X ) )
76a1i 11 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( y  e.  {
x  e.  ~P X  |  ( X  \  x )  ~<  X }  <->  ( y  C_  X  /\  ( X  \  y
)  ~<  X ) ) )
8 elex 3122 . . 3  |-  ( X  e.  dom  card  ->  X  e.  _V )
98adantr 465 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  e.  _V )
10 difid 3895 . . . 4  |-  ( X 
\  X )  =  (/)
11 infn0 7778 . . . . . 6  |-  ( om  ~<_  X  ->  X  =/=  (/) )
1211adantl 466 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  =/=  (/) )
13 0sdomg 7643 . . . . . 6  |-  ( X  e.  dom  card  ->  (
(/)  ~<  X  <->  X  =/=  (/) ) )
1413adantr 465 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( (/)  ~<  X  <->  X  =/=  (/) ) )
1512, 14mpbird 232 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  (/) 
~<  X )
1610, 15syl5eqbr 4480 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( X  \  X
)  ~<  X )
17 difeq2 3616 . . . . . 6  |-  ( y  =  X  ->  ( X  \  y )  =  ( X  \  X
) )
1817breq1d 4457 . . . . 5  |-  ( y  =  X  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
1918sbcieg 3364 . . . 4  |-  ( X  e.  dom  card  ->  (
[. X  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2019adantr 465 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( [. X  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2116, 20mpbird 232 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  [. X  /  y ]. ( X  \  y
)  ~<  X )
22 sdomirr 7651 . . 3  |-  -.  X  ~<  X
23 0ex 4577 . . . . 5  |-  (/)  e.  _V
24 difeq2 3616 . . . . . . 7  |-  ( y  =  (/)  ->  ( X 
\  y )  =  ( X  \  (/) ) )
25 dif0 3897 . . . . . . 7  |-  ( X 
\  (/) )  =  X
2624, 25syl6eq 2524 . . . . . 6  |-  ( y  =  (/)  ->  ( X 
\  y )  =  X )
2726breq1d 4457 . . . . 5  |-  ( y  =  (/)  ->  ( ( X  \  y ) 
~<  X  <->  X  ~<  X ) )
2823, 27sbcie 3366 . . . 4  |-  ( [. (/)  /  y ]. ( X  \  y )  ~<  X 
<->  X  ~<  X )
2928a1i 11 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( [. (/)  /  y ]. ( X  \  y
)  ~<  X  <->  X  ~<  X ) )
3022, 29mtbiri 303 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  -.  [. (/)  /  y ]. ( X  \  y
)  ~<  X )
31 simp1l 1020 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  X  e.  dom  card )
32 difexg 4595 . . . . . 6  |-  ( X  e.  dom  card  ->  ( X  \  w )  e.  _V )
3331, 32syl 16 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  w )  e. 
_V )
34 sscon 3638 . . . . . 6  |-  ( w 
C_  z  ->  ( X  \  z )  C_  ( X  \  w
) )
35343ad2ant3 1019 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  z )  C_  ( X  \  w
) )
36 ssdomg 7558 . . . . 5  |-  ( ( X  \  w )  e.  _V  ->  (
( X  \  z
)  C_  ( X  \  w )  ->  ( X  \  z )  ~<_  ( X  \  w ) ) )
3733, 35, 36sylc 60 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  z )  ~<_  ( X  \  w ) )
38 domsdomtr 7649 . . . . 5  |-  ( ( ( X  \  z
)  ~<_  ( X  \  w )  /\  ( X  \  w )  ~<  X )  ->  ( X  \  z )  ~<  X )
3938ex 434 . . . 4  |-  ( ( X  \  z )  ~<_  ( X  \  w
)  ->  ( ( X  \  w )  ~<  X  ->  ( X  \ 
z )  ~<  X ) )
4037, 39syl 16 . . 3  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  (
( X  \  w
)  ~<  X  ->  ( X  \  z )  ~<  X ) )
41 vex 3116 . . . 4  |-  w  e. 
_V
42 difeq2 3616 . . . . 5  |-  ( y  =  w  ->  ( X  \  y )  =  ( X  \  w
) )
4342breq1d 4457 . . . 4  |-  ( y  =  w  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  w )  ~<  X ) )
4441, 43sbcie 3366 . . 3  |-  ( [. w  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  w
)  ~<  X )
45 vex 3116 . . . 4  |-  z  e. 
_V
46 difeq2 3616 . . . . 5  |-  ( y  =  z  ->  ( X  \  y )  =  ( X  \  z
) )
4746breq1d 4457 . . . 4  |-  ( y  =  z  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  z )  ~<  X ) )
4845, 47sbcie 3366 . . 3  |-  ( [. z  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  z
)  ~<  X )
4940, 44, 483imtr4g 270 . 2  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( [. w  /  y ]. ( X  \  y
)  ~<  X  ->  [. z  /  y ]. ( X  \  y )  ~<  X ) )
50 infunsdom 8590 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( ( X  \ 
z )  ~<  X  /\  ( X  \  w
)  ~<  X ) )  ->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X )
5150ex 434 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( ( ( X 
\  z )  ~<  X  /\  ( X  \  w )  ~<  X )  ->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X ) )
52 difindi 3752 . . . . . 6  |-  ( X 
\  ( z  i^i  w ) )  =  ( ( X  \ 
z )  u.  ( X  \  w ) )
5352breq1i 4454 . . . . 5  |-  ( ( X  \  ( z  i^i  w ) ) 
~<  X  <->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X )
5451, 53syl6ibr 227 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( ( ( X 
\  z )  ~<  X  /\  ( X  \  w )  ~<  X )  ->  ( X  \ 
( z  i^i  w
) )  ~<  X ) )
55543ad2ant1 1017 . . 3  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  X )  ->  (
( ( X  \ 
z )  ~<  X  /\  ( X  \  w
)  ~<  X )  -> 
( X  \  (
z  i^i  w )
)  ~<  X ) )
5648, 44anbi12i 697 . . 3  |-  ( (
[. z  /  y ]. ( X  \  y
)  ~<  X  /\  [. w  /  y ]. ( X  \  y )  ~<  X )  <->  ( ( X  \  z )  ~<  X  /\  ( X  \  w )  ~<  X ) )
5745inex1 4588 . . . 4  |-  ( z  i^i  w )  e. 
_V
58 difeq2 3616 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( X  \  y )  =  ( X  \  (
z  i^i  w )
) )
5958breq1d 4457 . . . 4  |-  ( y  =  ( z  i^i  w )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X ) )
6057, 59sbcie 3366 . . 3  |-  ( [. ( z  i^i  w
)  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X )
6155, 56, 603imtr4g 270 . 2  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  X )  ->  (
( [. z  /  y ]. ( X  \  y
)  ~<  X  /\  [. w  /  y ]. ( X  \  y )  ~<  X )  ->  [. (
z  i^i  w )  /  y ]. ( X  \  y )  ~<  X ) )
627, 9, 21, 30, 49, 61isfild 20094 1  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  { x  e.  ~P X  |  ( X  \  x )  ~<  X }  e.  ( Fil `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113   [.wsbc 3331    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   class class class wbr 4447   dom cdm 4999   ` cfv 5586   omcom 6678    ~<_ cdom 7511    ~< csdm 7512   cardccrd 8312   Filcfil 20081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-oi 7931  df-card 8316  df-cda 8544  df-fbas 18187  df-fil 20082
This theorem is referenced by:  ufilen  20166
  Copyright terms: Public domain W3C validator