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Theorem csdfil 17605
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 3301 . . . . . 6  |-  ( x  =  y  ->  ( X  \  x )  =  ( X  \  y
) )
21breq1d 4049 . . . . 5  |-  ( x  =  y  ->  (
( X  \  x
)  ~<  X  <->  ( X  \  y )  ~<  X ) )
32elrab 2936 . . . 4  |-  ( y  e.  { x  e. 
~P X  |  ( X  \  x ) 
~<  X }  <->  ( y  e.  ~P X  /\  ( X  \  y )  ~<  X ) )
4 vex 2804 . . . . . 6  |-  y  e. 
_V
54elpw 3644 . . . . 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 2809 . . 3  |-  ( X  e.  dom  card  ->  X  e.  _V )
109adantr 451 . 2  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  e.  _V )
11 difid 3535 . . . 4  |-  ( X 
\  X )  =  (/)
12 infn0 7135 . . . . . 6  |-  ( om  ~<_  X  ->  X  =/=  (/) )
1312adantl 452 . . . . 5  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  X  =/=  (/) )
14 0sdomg 7006 . . . . . 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 4072 . . 3  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  -> 
( X  \  X
)  ~<  X )
18 difeq2 3301 . . . . . 6  |-  ( y  =  X  ->  ( X  \  y )  =  ( X  \  X
) )
1918breq1d 4049 . . . . 5  |-  ( y  =  X  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  X )  ~<  X ) )
2019sbcieg 3036 . . . 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 7014 . . 3  |-  -.  X  ~<  X
24 0ex 4166 . . . . 5  |-  (/)  e.  _V
25 difeq2 3301 . . . . . . 7  |-  ( y  =  (/)  ->  ( X 
\  y )  =  ( X  \  (/) ) )
26 dif0 3537 . . . . . . 7  |-  ( X 
\  (/) )  =  X
2725, 26syl6eq 2344 . . . . . 6  |-  ( y  =  (/)  ->  ( X 
\  y )  =  X )
2827breq1d 4049 . . . . 5  |-  ( y  =  (/)  ->  ( ( X  \  y ) 
~<  X  <->  X  ~<  X ) )
2924, 28sbcie 3038 . . . 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 4178 . . . . . 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 3323 . . . . . 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 6923 . . . . 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 7012 . . . . 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 2804 . . . 4  |-  w  e. 
_V
43 difeq2 3301 . . . . 5  |-  ( y  =  w  ->  ( X  \  y )  =  ( X  \  w
) )
4443breq1d 4049 . . . 4  |-  ( y  =  w  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  w )  ~<  X ) )
4542, 44sbcie 3038 . . 3  |-  ( [. w  /  y ]. ( X  \  y )  ~<  X 
<->  ( X  \  w
)  ~<  X )
46 vex 2804 . . . 4  |-  z  e. 
_V
47 difeq2 3301 . . . . 5  |-  ( y  =  z  ->  ( X  \  y )  =  ( X  \  z
) )
4847breq1d 4049 . . . 4  |-  ( y  =  z  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  z )  ~<  X ) )
4946, 48sbcie 3038 . . 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 7856 . . . . . 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 3436 . . . . . 6  |-  ( X 
\  ( z  i^i  w ) )  =  ( ( X  \ 
z )  u.  ( X  \  w ) )
5453breq1i 4046 . . . . 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 4171 . . . 4  |-  ( z  i^i  w )  e. 
_V
59 difeq2 3301 . . . . 5  |-  ( y  =  ( z  i^i  w )  ->  ( X  \  y )  =  ( X  \  (
z  i^i  w )
) )
6059breq1d 4049 . . . 4  |-  ( y  =  ( z  i^i  w )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( z  i^i  w
) )  ~<  X ) )
6158, 60sbcie 3038 . . 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 17569 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 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801   [.wsbc 3004    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   class class class wbr 4039   omcom 4672   dom cdm 4705   ` cfv 5271    ~<_ cdom 6877    ~< csdm 6878   cardccrd 7584   Filcfil 17556
This theorem is referenced by:  ufilen  17641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-cda 7810  df-fbas 17536  df-fil 17557
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