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Theorem csdfil 19592
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 3569 . . . . . 6  |-  ( x  =  y  ->  ( X  \  x )  =  ( X  \  y
) )
21breq1d 4403 . . . . 5  |-  ( x  =  y  ->  (
( X  \  x
)  ~<  X  <->  ( X  \  y )  ~<  X ) )
32elrab 3217 . . . 4  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  e.  ~P X  /\  ( X  \  y )  ~<  X ) )
4 selpw 3968 . . . . 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 3080 . . 3  |-  ( X  e.  dom  card  ->  X  e.  _V )
98adantr 465 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  e.  _V )
10 difid 3848 . . . 4  |-  ( X 
\  X )  =  (/)
11 infn0 7678 . . . . . 6  |-  ( om  ~<_  X  ->  X  =/=  (/) )
1211adantl 466 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  =/=  (/) )
13 0sdomg 7543 . . . . . 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 4426 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( X  \  X
)  ~<  X )
17 difeq2 3569 . . . . . 6  |-  ( y  =  X  ->  ( X  \  y )  =  ( X  \  X
) )
1817breq1d 4403 . . . . 5  |-  ( y  =  X  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
1918sbcieg 3320 . . . 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 7551 . . 3  |-  -.  X  ~<  X
23 0ex 4523 . . . . 5  |-  (/)  e.  _V
24 difeq2 3569 . . . . . . 7  |-  ( y  =  (/)  ->  ( X 
\  y )  =  ( X  \  (/) ) )
25 dif0 3850 . . . . . . 7  |-  ( X 
\  (/) )  =  X
2624, 25syl6eq 2508 . . . . . 6  |-  ( y  =  (/)  ->  ( X 
\  y )  =  X )
2726breq1d 4403 . . . . 5  |-  ( y  =  (/)  ->  ( ( X  \  y ) 
~<  X  <->  X  ~<  X ) )
2823, 27sbcie 3322 . . . 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 1012 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  X  e.  dom  card )
32 difexg 4541 . . . . . 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 3591 . . . . . 6  |-  ( w 
C_  z  ->  ( X  \  z )  C_  ( X  \  w
) )
35343ad2ant3 1011 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  z )  C_  ( X  \  w
) )
36 ssdomg 7458 . . . . 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 7549 . . . . 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 3074 . . . 4  |-  w  e. 
_V
42 difeq2 3569 . . . . 5  |-  ( y  =  w  ->  ( X  \  y )  =  ( X  \  w
) )
4342breq1d 4403 . . . 4  |-  ( y  =  w  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  w )  ~<  X ) )
4441, 43sbcie 3322 . . 3  |-  ( [. w  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  w
)  ~<  X )
45 vex 3074 . . . 4  |-  z  e. 
_V
46 difeq2 3569 . . . . 5  |-  ( y  =  z  ->  ( X  \  y )  =  ( X  \  z
) )
4746breq1d 4403 . . . 4  |-  ( y  =  z  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  z )  ~<  X ) )
4845, 47sbcie 3322 . . 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 8487 . . . . . 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 3705 . . . . . 6  |-  ( X 
\  ( z  i^i  w ) )  =  ( ( X  \ 
z )  u.  ( X  \  w ) )
5352breq1i 4400 . . . . 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 1009 . . 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 4534 . . . 4  |-  ( z  i^i  w )  e. 
_V
58 difeq2 3569 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( X  \  y )  =  ( X  \  (
z  i^i  w )
) )
5958breq1d 4403 . . . 4  |-  ( y  =  ( z  i^i  w )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X ) )
6057, 59sbcie 3322 . . 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 19556 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   {crab 2799   _Vcvv 3071   [.wsbc 3287    \ cdif 3426    u. cun 3427    i^i cin 3428    C_ wss 3429   (/)c0 3738   ~Pcpw 3961   class class class wbr 4393   dom cdm 4941   ` cfv 5519   omcom 6579    ~<_ cdom 7411    ~< csdm 7412   cardccrd 8209   Filcfil 19543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-oi 7828  df-card 8213  df-cda 8441  df-fbas 17932  df-fil 19544
This theorem is referenced by:  ufilen  19628
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