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Theorem isfild 19431
Description: Sufficient condition for a set of the form  { x  e.  ~P A  |  ph } to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Hypotheses
Ref Expression
isfild.1  |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )
isfild.2  |-  ( ph  ->  A  e.  _V )
isfild.3  |-  ( ph  ->  [. A  /  x ]. ps )
isfild.4  |-  ( ph  ->  -.  [. (/)  /  x ]. ps )
isfild.5  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( [. z  /  x ]. ps  ->  [. y  /  x ]. ps ) )
isfild.6  |-  ( (
ph  /\  y  C_  A  /\  z  C_  A
)  ->  ( ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )  ->  [. ( y  i^i  z )  /  x ]. ps ) )
Assertion
Ref Expression
isfild  |-  ( ph  ->  F  e.  ( Fil `  A ) )
Distinct variable groups:    x, y, A    z, A    x, F, y    y, z, F    ph, x, y    ph, z    ps, y
Allowed substitution hints:    ps( x, z)

Proof of Theorem isfild
StepHypRef Expression
1 isfild.1 . . . . 5  |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )
2 selpw 3867 . . . . . . 7  |-  ( x  e.  ~P A  <->  x  C_  A
)
32biimpri 206 . . . . . 6  |-  ( x 
C_  A  ->  x  e.  ~P A )
43adantr 465 . . . . 5  |-  ( ( x  C_  A  /\  ps )  ->  x  e. 
~P A )
51, 4syl6bi 228 . . . 4  |-  ( ph  ->  ( x  e.  F  ->  x  e.  ~P A
) )
65ssrdv 3362 . . 3  |-  ( ph  ->  F  C_  ~P A
)
7 isfild.4 . . . 4  |-  ( ph  ->  -.  [. (/)  /  x ]. ps )
8 isfild.2 . . . . . 6  |-  ( ph  ->  A  e.  _V )
91, 8isfildlem 19430 . . . . 5  |-  ( ph  ->  ( (/)  e.  F  <->  (
(/)  C_  A  /\  [. (/)  /  x ]. ps ) ) )
10 simpr 461 . . . . 5  |-  ( (
(/)  C_  A  /\  [. (/)  /  x ]. ps )  ->  [. (/)  /  x ]. ps )
119, 10syl6bi 228 . . . 4  |-  ( ph  ->  ( (/)  e.  F  ->  [. (/)  /  x ]. ps ) )
127, 11mtod 177 . . 3  |-  ( ph  ->  -.  (/)  e.  F )
13 isfild.3 . . . . 5  |-  ( ph  ->  [. A  /  x ]. ps )
14 ssid 3375 . . . . 5  |-  A  C_  A
1513, 14jctil 537 . . . 4  |-  ( ph  ->  ( A  C_  A  /\  [. A  /  x ]. ps ) )
161, 8isfildlem 19430 . . . 4  |-  ( ph  ->  ( A  e.  F  <->  ( A  C_  A  /\  [. A  /  x ]. ps ) ) )
1715, 16mpbird 232 . . 3  |-  ( ph  ->  A  e.  F )
186, 12, 173jca 1168 . 2  |-  ( ph  ->  ( F  C_  ~P A  /\  -.  (/)  e.  F  /\  A  e.  F
) )
19 elpwi 3869 . . . 4  |-  ( y  e.  ~P A  -> 
y  C_  A )
20 isfild.5 . . . . . . . . . . 11  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( [. z  /  x ]. ps  ->  [. y  /  x ]. ps ) )
21 simp2 989 . . . . . . . . . . 11  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  y  C_  A )
2220, 21jctild 543 . . . . . . . . . 10  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( [. z  /  x ]. ps  ->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2322adantld 467 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( (
z  C_  A  /\  [. z  /  x ]. ps )  ->  ( y 
C_  A  /\  [. y  /  x ]. ps )
) )
241, 8isfildlem 19430 . . . . . . . . . 10  |-  ( ph  ->  ( z  e.  F  <->  ( z  C_  A  /\  [. z  /  x ]. ps ) ) )
25243ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( z  e.  F  <->  ( z  C_  A  /\  [. z  /  x ]. ps ) ) )
261, 8isfildlem 19430 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
27263ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2823, 25, 273imtr4d 268 . . . . . . . 8  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( z  e.  F  ->  y  e.  F ) )
29283expa 1187 . . . . . . 7  |-  ( ( ( ph  /\  y  C_  A )  /\  z  C_  y )  ->  (
z  e.  F  -> 
y  e.  F ) )
3029impancom 440 . . . . . 6  |-  ( ( ( ph  /\  y  C_  A )  /\  z  e.  F )  ->  (
z  C_  y  ->  y  e.  F ) )
3130rexlimdva 2841 . . . . 5  |-  ( (
ph  /\  y  C_  A )  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F ) )
3231ex 434 . . . 4  |-  ( ph  ->  ( y  C_  A  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F ) ) )
3319, 32syl5 32 . . 3  |-  ( ph  ->  ( y  e.  ~P A  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F
) ) )
3433ralrimiv 2798 . 2  |-  ( ph  ->  A. y  e.  ~P  A ( E. z  e.  F  z  C_  y  ->  y  e.  F
) )
35 ssinss1 3578 . . . . . . 7  |-  ( y 
C_  A  ->  (
y  i^i  z )  C_  A )
3635ad2antrr 725 . . . . . 6  |-  ( ( ( y  C_  A  /\  [. y  /  x ]. ps )  /\  (
z  C_  A  /\  [. z  /  x ]. ps ) )  ->  (
y  i^i  z )  C_  A )
3736a1i 11 . . . . 5  |-  ( ph  ->  ( ( ( y 
C_  A  /\  [. y  /  x ]. ps )  /\  ( z  C_  A  /\  [. z  /  x ]. ps ) )  -> 
( y  i^i  z
)  C_  A )
)
38 an4 820 . . . . . 6  |-  ( ( ( y  C_  A  /\  [. y  /  x ]. ps )  /\  (
z  C_  A  /\  [. z  /  x ]. ps ) )  <->  ( (
y  C_  A  /\  z  C_  A )  /\  ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )
) )
39 isfild.6 . . . . . . . 8  |-  ( (
ph  /\  y  C_  A  /\  z  C_  A
)  ->  ( ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )  ->  [. ( y  i^i  z )  /  x ]. ps ) )
40393expb 1188 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  A  /\  z  C_  A ) )  -> 
( ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )  ->  [. (
y  i^i  z )  /  x ]. ps )
)
4140expimpd 603 . . . . . 6  |-  ( ph  ->  ( ( ( y 
C_  A  /\  z  C_  A )  /\  ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )
)  ->  [. ( y  i^i  z )  /  x ]. ps ) )
4238, 41syl5bi 217 . . . . 5  |-  ( ph  ->  ( ( ( y 
C_  A  /\  [. y  /  x ]. ps )  /\  ( z  C_  A  /\  [. z  /  x ]. ps ) )  ->  [. ( y  i^i  z
)  /  x ]. ps ) )
4337, 42jcad 533 . . . 4  |-  ( ph  ->  ( ( ( y 
C_  A  /\  [. y  /  x ]. ps )  /\  ( z  C_  A  /\  [. z  /  x ]. ps ) )  -> 
( ( y  i^i  z )  C_  A  /\  [. ( y  i^i  z )  /  x ]. ps ) ) )
4426, 24anbi12d 710 . . . 4  |-  ( ph  ->  ( ( y  e.  F  /\  z  e.  F )  <->  ( (
y  C_  A  /\  [. y  /  x ]. ps )  /\  (
z  C_  A  /\  [. z  /  x ]. ps ) ) ) )
451, 8isfildlem 19430 . . . 4  |-  ( ph  ->  ( ( y  i^i  z )  e.  F  <->  ( ( y  i^i  z
)  C_  A  /\  [. ( y  i^i  z
)  /  x ]. ps ) ) )
4643, 44, 453imtr4d 268 . . 3  |-  ( ph  ->  ( ( y  e.  F  /\  z  e.  F )  ->  (
y  i^i  z )  e.  F ) )
4746ralrimivv 2807 . 2  |-  ( ph  ->  A. y  e.  F  A. z  e.  F  ( y  i^i  z
)  e.  F )
48 isfil2 19429 . 2  |-  ( F  e.  ( Fil `  A
)  <->  ( ( F 
C_  ~P A  /\  -.  (/) 
e.  F  /\  A  e.  F )  /\  A. y  e.  ~P  A
( E. z  e.  F  z  C_  y  ->  y  e.  F )  /\  A. y  e.  F  A. z  e.  F  ( y  i^i  z )  e.  F
) )
4918, 34, 47, 48syl3anbrc 1172 1  |-  ( ph  ->  F  e.  ( Fil `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972   [.wsbc 3186    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   ` cfv 5418   Filcfil 19418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-fbas 17814  df-fil 19419
This theorem is referenced by:  snfil  19437  fgcl  19451  filuni  19458  cfinfil  19466  csdfil  19467  supfil  19468  fin1aufil  19505
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