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Theorem csdfil 17918
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 3451 . . . . . 6  |-  ( x  =  y  ->  ( X  \  x )  =  ( X  \  y
) )
21breq1d 4214 . . . . 5  |-  ( x  =  y  ->  (
( X  \  x
)  ~<  X  <->  ( X  \  y )  ~<  X ) )
32elrab 3084 . . . 4  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  e.  ~P X  /\  ( X  \  y )  ~<  X ) )
4 vex 2951 . . . . . 6  |-  y  e. 
_V
54elpw 3797 . . . . 5  |-  ( y  e.  ~P X  <->  y  C_  X )
65anbi1i 677 . . . 4  |-  ( ( y  e.  ~P X  /\  ( X  \  y
)  ~<  X )  <->  ( y  C_  X  /\  ( X 
\  y )  ~<  X ) )
73, 6bitri 241 . . 3  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  C_  X  /\  ( X 
\  y )  ~<  X ) )
87a1i 11 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( y  e.  {
x  e.  ~P X  |  ( X  \  x )  ~<  X }  <->  ( y  C_  X  /\  ( X  \  y
)  ~<  X ) ) )
9 elex 2956 . . 3  |-  ( X  e.  dom  card  ->  X  e.  _V )
109adantr 452 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  e.  _V )
11 difid 3688 . . . 4  |-  ( X 
\  X )  =  (/)
12 infn0 7361 . . . . . 6  |-  ( om  ~<_  X  ->  X  =/=  (/) )
1312adantl 453 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  =/=  (/) )
14 0sdomg 7228 . . . . . 6  |-  ( X  e.  dom  card  ->  (
(/)  ~<  X  <->  X  =/=  (/) ) )
1514adantr 452 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( (/)  ~<  X  <->  X  =/=  (/) ) )
1613, 15mpbird 224 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  (/) 
~<  X )
1711, 16syl5eqbr 4237 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( X  \  X
)  ~<  X )
18 difeq2 3451 . . . . . 6  |-  ( y  =  X  ->  ( X  \  y )  =  ( X  \  X
) )
1918breq1d 4214 . . . . 5  |-  ( y  =  X  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2019sbcieg 3185 . . . 4  |-  ( X  e.  dom  card  ->  (
[. X  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2120adantr 452 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( [. X  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2217, 21mpbird 224 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  [. X  /  y ]. ( X  \  y
)  ~<  X )
23 sdomirr 7236 . . 3  |-  -.  X  ~<  X
24 0ex 4331 . . . . 5  |-  (/)  e.  _V
25 difeq2 3451 . . . . . . 7  |-  ( y  =  (/)  ->  ( X 
\  y )  =  ( X  \  (/) ) )
26 dif0 3690 . . . . . . 7  |-  ( X 
\  (/) )  =  X
2725, 26syl6eq 2483 . . . . . 6  |-  ( y  =  (/)  ->  ( X 
\  y )  =  X )
2827breq1d 4214 . . . . 5  |-  ( y  =  (/)  ->  ( ( X  \  y ) 
~<  X  <->  X  ~<  X ) )
2924, 28sbcie 3187 . . . 4  |-  ( [. (/)  /  y ]. ( X  \  y )  ~<  X 
<->  X  ~<  X )
3029a1i 11 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( [. (/)  /  y ]. ( X  \  y
)  ~<  X  <->  X  ~<  X ) )
3123, 30mtbiri 295 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  -.  [. (/)  /  y ]. ( X  \  y
)  ~<  X )
32 simp1l 981 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  X  e.  dom  card )
33 difexg 4343 . . . . . 6  |-  ( X  e.  dom  card  ->  ( X  \  w )  e.  _V )
3432, 33syl 16 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  w )  e. 
_V )
35 sscon 3473 . . . . . 6  |-  ( w 
C_  z  ->  ( X  \  z )  C_  ( X  \  w
) )
36353ad2ant3 980 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  z )  C_  ( X  \  w
) )
37 ssdomg 7145 . . . . 5  |-  ( ( X  \  w )  e.  _V  ->  (
( X  \  z
)  C_  ( X  \  w )  ->  ( X  \  z )  ~<_  ( X  \  w ) ) )
3834, 36, 37sylc 58 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  z )  ~<_  ( X  \  w ) )
39 domsdomtr 7234 . . . . 5  |-  ( ( ( X  \  z
)  ~<_  ( X  \  w )  /\  ( X  \  w )  ~<  X )  ->  ( X  \  z )  ~<  X )
4039ex 424 . . . 4  |-  ( ( X  \  z )  ~<_  ( X  \  w
)  ->  ( ( X  \  w )  ~<  X  ->  ( X  \ 
z )  ~<  X ) )
4138, 40syl 16 . . 3  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  (
( X  \  w
)  ~<  X  ->  ( X  \  z )  ~<  X ) )
42 vex 2951 . . . 4  |-  w  e. 
_V
43 difeq2 3451 . . . . 5  |-  ( y  =  w  ->  ( X  \  y )  =  ( X  \  w
) )
4443breq1d 4214 . . . 4  |-  ( y  =  w  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  w )  ~<  X ) )
4542, 44sbcie 3187 . . 3  |-  ( [. w  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  w
)  ~<  X )
46 vex 2951 . . . 4  |-  z  e. 
_V
47 difeq2 3451 . . . . 5  |-  ( y  =  z  ->  ( X  \  y )  =  ( X  \  z
) )
4847breq1d 4214 . . . 4  |-  ( y  =  z  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  z )  ~<  X ) )
4946, 48sbcie 3187 . . 3  |-  ( [. z  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  z
)  ~<  X )
5041, 45, 493imtr4g 262 . 2  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( [. w  /  y ]. ( X  \  y
)  ~<  X  ->  [. z  /  y ]. ( X  \  y )  ~<  X ) )
51 infunsdom 8086 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( ( X  \ 
z )  ~<  X  /\  ( X  \  w
)  ~<  X ) )  ->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X )
5251ex 424 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( ( ( X 
\  z )  ~<  X  /\  ( X  \  w )  ~<  X )  ->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X ) )
53 difindi 3587 . . . . . 6  |-  ( X 
\  ( z  i^i  w ) )  =  ( ( X  \ 
z )  u.  ( X  \  w ) )
5453breq1i 4211 . . . . 5  |-  ( ( X  \  ( z  i^i  w ) ) 
~<  X  <->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X )
5552, 54syl6ibr 219 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( ( ( X 
\  z )  ~<  X  /\  ( X  \  w )  ~<  X )  ->  ( X  \ 
( z  i^i  w
) )  ~<  X ) )
56553ad2ant1 978 . . 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 ) )
5749, 45anbi12i 679 . . 3  |-  ( (
[. z  /  y ]. ( X  \  y
)  ~<  X  /\  [. w  /  y ]. ( X  \  y )  ~<  X )  <->  ( ( X  \  z )  ~<  X  /\  ( X  \  w )  ~<  X ) )
5846inex1 4336 . . . 4  |-  ( z  i^i  w )  e. 
_V
59 difeq2 3451 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( X  \  y )  =  ( X  \  (
z  i^i  w )
) )
6059breq1d 4214 . . . 4  |-  ( y  =  ( z  i^i  w )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X ) )
6158, 60sbcie 3187 . . 3  |-  ( [. ( z  i^i  w
)  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X )
6256, 57, 613imtr4g 262 . 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 ) )
638, 10, 22, 31, 50, 62isfild 17882 1  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  { x  e.  ~P X  |  ( X  \  x )  ~<  X }  e.  ( Fil `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948   [.wsbc 3153    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   class class class wbr 4204   omcom 4837   dom cdm 4870   ` cfv 5446    ~<_ cdom 7099    ~< csdm 7100   cardccrd 7814   Filcfil 17869
This theorem is referenced by:  ufilen  17954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-cda 8040  df-fbas 16691  df-fil 17870
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