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Theorem isfild 17569
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 vex 2804 . . . . . . . 8  |-  x  e. 
_V
32elpw 3644 . . . . . . 7  |-  ( x  e.  ~P A  <->  x  C_  A
)
43biimpri 197 . . . . . 6  |-  ( x 
C_  A  ->  x  e.  ~P A )
54adantr 451 . . . . 5  |-  ( ( x  C_  A  /\  ps )  ->  x  e. 
~P A )
61, 5syl6bi 219 . . . 4  |-  ( ph  ->  ( x  e.  F  ->  x  e.  ~P A
) )
76ssrdv 3198 . . 3  |-  ( ph  ->  F  C_  ~P A
)
8 isfild.4 . . . 4  |-  ( ph  ->  -.  [. (/)  /  x ]. ps )
9 isfild.2 . . . . . 6  |-  ( ph  ->  A  e.  _V )
101, 9isfildlem 17568 . . . . 5  |-  ( ph  ->  ( (/)  e.  F  <->  (
(/)  C_  A  /\  [. (/)  /  x ]. ps ) ) )
11 simpr 447 . . . . 5  |-  ( (
(/)  C_  A  /\  [. (/)  /  x ]. ps )  ->  [. (/)  /  x ]. ps )
1210, 11syl6bi 219 . . . 4  |-  ( ph  ->  ( (/)  e.  F  ->  [. (/)  /  x ]. ps ) )
138, 12mtod 168 . . 3  |-  ( ph  ->  -.  (/)  e.  F )
14 isfild.3 . . . . 5  |-  ( ph  ->  [. A  /  x ]. ps )
15 ssid 3210 . . . . 5  |-  A  C_  A
1614, 15jctil 523 . . . 4  |-  ( ph  ->  ( A  C_  A  /\  [. A  /  x ]. ps ) )
171, 9isfildlem 17568 . . . 4  |-  ( ph  ->  ( A  e.  F  <->  ( A  C_  A  /\  [. A  /  x ]. ps ) ) )
1816, 17mpbird 223 . . 3  |-  ( ph  ->  A  e.  F )
197, 13, 183jca 1132 . 2  |-  ( ph  ->  ( F  C_  ~P A  /\  -.  (/)  e.  F  /\  A  e.  F
) )
20 elpwi 3646 . . . 4  |-  ( y  e.  ~P A  -> 
y  C_  A )
21 isfild.5 . . . . . . . . . . 11  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( [. z  /  x ]. ps  ->  [. y  /  x ]. ps ) )
22 simp2 956 . . . . . . . . . . 11  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  y  C_  A )
2321, 22jctild 527 . . . . . . . . . 10  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( [. z  /  x ]. ps  ->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2423adantld 453 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( (
z  C_  A  /\  [. z  /  x ]. ps )  ->  ( y 
C_  A  /\  [. y  /  x ]. ps )
) )
251, 9isfildlem 17568 . . . . . . . . . 10  |-  ( ph  ->  ( z  e.  F  <->  ( z  C_  A  /\  [. z  /  x ]. ps ) ) )
26253ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( z  e.  F  <->  ( z  C_  A  /\  [. z  /  x ]. ps ) ) )
271, 9isfildlem 17568 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
28273ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2924, 26, 283imtr4d 259 . . . . . . . 8  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( z  e.  F  ->  y  e.  F ) )
30293expa 1151 . . . . . . 7  |-  ( ( ( ph  /\  y  C_  A )  /\  z  C_  y )  ->  (
z  e.  F  -> 
y  e.  F ) )
3130impancom 427 . . . . . 6  |-  ( ( ( ph  /\  y  C_  A )  /\  z  e.  F )  ->  (
z  C_  y  ->  y  e.  F ) )
3231rexlimdva 2680 . . . . 5  |-  ( (
ph  /\  y  C_  A )  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F ) )
3332ex 423 . . . 4  |-  ( ph  ->  ( y  C_  A  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F ) ) )
3420, 33syl5 28 . . 3  |-  ( ph  ->  ( y  e.  ~P A  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F
) ) )
3534ralrimiv 2638 . 2  |-  ( ph  ->  A. y  e.  ~P  A ( E. z  e.  F  z  C_  y  ->  y  e.  F
) )
36 ssinss1 3410 . . . . . . 7  |-  ( y 
C_  A  ->  (
y  i^i  z )  C_  A )
3736ad2antrr 706 . . . . . 6  |-  ( ( ( y  C_  A  /\  [. y  /  x ]. ps )  /\  (
z  C_  A  /\  [. z  /  x ]. ps ) )  ->  (
y  i^i  z )  C_  A )
3837a1i 10 . . . . 5  |-  ( ph  ->  ( ( ( y 
C_  A  /\  [. y  /  x ]. ps )  /\  ( z  C_  A  /\  [. z  /  x ]. ps ) )  -> 
( y  i^i  z
)  C_  A )
)
39 an4 797 . . . . . 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 )
) )
40 isfild.6 . . . . . . . 8  |-  ( (
ph  /\  y  C_  A  /\  z  C_  A
)  ->  ( ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )  ->  [. ( y  i^i  z )  /  x ]. ps ) )
41403expb 1152 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  A  /\  z  C_  A ) )  -> 
( ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )  ->  [. (
y  i^i  z )  /  x ]. ps )
)
4241expimpd 586 . . . . . 6  |-  ( ph  ->  ( ( ( y 
C_  A  /\  z  C_  A )  /\  ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )
)  ->  [. ( y  i^i  z )  /  x ]. ps ) )
4339, 42syl5bi 208 . . . . 5  |-  ( ph  ->  ( ( ( y 
C_  A  /\  [. y  /  x ]. ps )  /\  ( z  C_  A  /\  [. z  /  x ]. ps ) )  ->  [. ( y  i^i  z
)  /  x ]. ps ) )
4438, 43jcad 519 . . . 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 ) ) )
4527, 25anbi12d 691 . . . 4  |-  ( ph  ->  ( ( y  e.  F  /\  z  e.  F )  <->  ( (
y  C_  A  /\  [. y  /  x ]. ps )  /\  (
z  C_  A  /\  [. z  /  x ]. ps ) ) ) )
461, 9isfildlem 17568 . . . 4  |-  ( ph  ->  ( ( y  i^i  z )  e.  F  <->  ( ( y  i^i  z
)  C_  A  /\  [. ( y  i^i  z
)  /  x ]. ps ) ) )
4744, 45, 463imtr4d 259 . . 3  |-  ( ph  ->  ( ( y  e.  F  /\  z  e.  F )  ->  (
y  i^i  z )  e.  F ) )
4847ralrimivv 2647 . 2  |-  ( ph  ->  A. y  e.  F  A. z  e.  F  ( y  i^i  z
)  e.  F )
49 isfil2 17567 . 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
) )
5019, 35, 48, 49syl3anbrc 1136 1  |-  ( ph  ->  F  e.  ( Fil `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   [.wsbc 3004    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   ` cfv 5271   Filcfil 17556
This theorem is referenced by:  snfil  17575  fgcl  17589  filuni  17596  cfinfil  17604  csdfil  17605  supfil  17606  fin1aufil  17643
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-fv 5279  df-fbas 17536  df-fil 17557
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