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Theorem csdfil 17589
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 3288 . . . . . 6  |-  ( x  =  y  ->  ( X  \  x )  =  ( X  \  y
) )
21breq1d 4033 . . . . 5  |-  ( x  =  y  ->  (
( X  \  x
)  ~<  X  <->  ( X  \  y )  ~<  X ) )
32elrab 2923 . . . 4  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  e.  ~P X  /\  ( X  \  y )  ~<  X ) )
4 vex 2791 . . . . . 6  |-  y  e. 
_V
54elpw 3631 . . . . 5  |-  ( y  e.  ~P X  <->  y  C_  X )
65anbi1i 676 . . . 4  |-  ( ( y  e.  ~P X  /\  ( X  \  y
)  ~<  X )  <->  ( y  C_  X  /\  ( X 
\  y )  ~<  X ) )
73, 6bitri 240 . . 3  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  C_  X  /\  ( X 
\  y )  ~<  X ) )
87a1i 10 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( y  e.  {
x  e.  ~P X  |  ( X  \  x )  ~<  X }  <->  ( y  C_  X  /\  ( X  \  y
)  ~<  X ) ) )
9 elex 2796 . . 3  |-  ( X  e.  dom  card  ->  X  e.  _V )
109adantr 451 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  e.  _V )
11 difid 3522 . . . 4  |-  ( X 
\  X )  =  (/)
12 infn0 7119 . . . . . 6  |-  ( om  ~<_  X  ->  X  =/=  (/) )
1312adantl 452 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  =/=  (/) )
14 0sdomg 6990 . . . . . 6  |-  ( X  e.  dom  card  ->  (
(/)  ~<  X  <->  X  =/=  (/) ) )
1514adantr 451 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( (/)  ~<  X  <->  X  =/=  (/) ) )
1613, 15mpbird 223 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  (/) 
~<  X )
1711, 16syl5eqbr 4056 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( X  \  X
)  ~<  X )
18 difeq2 3288 . . . . . 6  |-  ( y  =  X  ->  ( X  \  y )  =  ( X  \  X
) )
1918breq1d 4033 . . . . 5  |-  ( y  =  X  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2019sbcieg 3023 . . . 4  |-  ( X  e.  dom  card  ->  (
[. X  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2120adantr 451 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( [. X  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2217, 21mpbird 223 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  [. X  /  y ]. ( X  \  y
)  ~<  X )
23 sdomirr 6998 . . 3  |-  -.  X  ~<  X
24 0ex 4150 . . . . 5  |-  (/)  e.  _V
25 difeq2 3288 . . . . . . 7  |-  ( y  =  (/)  ->  ( X 
\  y )  =  ( X  \  (/) ) )
26 dif0 3524 . . . . . . 7  |-  ( X 
\  (/) )  =  X
2725, 26syl6eq 2331 . . . . . 6  |-  ( y  =  (/)  ->  ( X 
\  y )  =  X )
2827breq1d 4033 . . . . 5  |-  ( y  =  (/)  ->  ( ( X  \  y ) 
~<  X  <->  X  ~<  X ) )
2924, 28sbcie 3025 . . . 4  |-  ( [. (/)  /  y ]. ( X  \  y )  ~<  X 
<->  X  ~<  X )
3029a1i 10 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( [. (/)  /  y ]. ( X  \  y
)  ~<  X  <->  X  ~<  X ) )
3123, 30mtbiri 294 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  -.  [. (/)  /  y ]. ( X  \  y
)  ~<  X )
32 simp1l 979 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  X  e.  dom  card )
33 difexg 4162 . . . . . 6  |-  ( X  e.  dom  card  ->  ( X  \  w )  e.  _V )
3432, 33syl 15 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  w )  e. 
_V )
35 sscon 3310 . . . . . 6  |-  ( w 
C_  z  ->  ( X  \  z )  C_  ( X  \  w
) )
36353ad2ant3 978 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  z )  C_  ( X  \  w
) )
37 ssdomg 6907 . . . . 5  |-  ( ( X  \  w )  e.  _V  ->  (
( X  \  z
)  C_  ( X  \  w )  ->  ( X  \  z )  ~<_  ( X  \  w ) ) )
3834, 36, 37sylc 56 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  ( X  \  z )  ~<_  ( X  \  w ) )
39 domsdomtr 6996 . . . . 5  |-  ( ( ( X  \  z
)  ~<_  ( X  \  w )  /\  ( X  \  w )  ~<  X )  ->  ( X  \  z )  ~<  X )
4039ex 423 . . . 4  |-  ( ( X  \  z )  ~<_  ( X  \  w
)  ->  ( ( X  \  w )  ~<  X  ->  ( X  \ 
z )  ~<  X ) )
4138, 40syl 15 . . 3  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  z  C_  X  /\  w  C_  z )  ->  (
( X  \  w
)  ~<  X  ->  ( X  \  z )  ~<  X ) )
42 vex 2791 . . . 4  |-  w  e. 
_V
43 difeq2 3288 . . . . 5  |-  ( y  =  w  ->  ( X  \  y )  =  ( X  \  w
) )
4443breq1d 4033 . . . 4  |-  ( y  =  w  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  w )  ~<  X ) )
4542, 44sbcie 3025 . . 3  |-  ( [. w  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  w
)  ~<  X )
46 vex 2791 . . . 4  |-  z  e. 
_V
47 difeq2 3288 . . . . 5  |-  ( y  =  z  ->  ( X  \  y )  =  ( X  \  z
) )
4847breq1d 4033 . . . 4  |-  ( y  =  z  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  z )  ~<  X ) )
4946, 48sbcie 3025 . . 3  |-  ( [. z  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  z
)  ~<  X )
5041, 45, 493imtr4g 261 . 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 7840 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( ( X  \ 
z )  ~<  X  /\  ( X  \  w
)  ~<  X ) )  ->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X )
5251ex 423 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( ( ( X 
\  z )  ~<  X  /\  ( X  \  w )  ~<  X )  ->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X ) )
53 difindi 3423 . . . . . 6  |-  ( X 
\  ( z  i^i  w ) )  =  ( ( X  \ 
z )  u.  ( X  \  w ) )
5453breq1i 4030 . . . . 5  |-  ( ( X  \  ( z  i^i  w ) ) 
~<  X  <->  ( ( X 
\  z )  u.  ( X  \  w
) )  ~<  X )
5552, 54syl6ibr 218 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( ( ( X 
\  z )  ~<  X  /\  ( X  \  w )  ~<  X )  ->  ( X  \ 
( z  i^i  w
) )  ~<  X ) )
56553ad2ant1 976 . . 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 678 . . 3  |-  ( (
[. z  /  y ]. ( X  \  y
)  ~<  X  /\  [. w  /  y ]. ( X  \  y )  ~<  X )  <->  ( ( X  \  z )  ~<  X  /\  ( X  \  w )  ~<  X ) )
5846inex1 4155 . . . 4  |-  ( z  i^i  w )  e. 
_V
59 difeq2 3288 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( X  \  y )  =  ( X  \  (
z  i^i  w )
) )
6059breq1d 4033 . . . 4  |-  ( y  =  ( z  i^i  w )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X ) )
6158, 60sbcie 3025 . . 3  |-  ( [. ( z  i^i  w
)  /  y ]. ( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X )
6256, 57, 613imtr4g 261 . 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 17553 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023   omcom 4656   dom cdm 4689   ` cfv 5255    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568   Filcfil 17540
This theorem is referenced by:  ufilen  17625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-cda 7794  df-fbas 17520  df-fil 17541
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