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Theorem isfild 20804
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 3992 . . . . . . 7  |-  ( x  e.  ~P A  <->  x  C_  A
)
32biimpri 209 . . . . . 6  |-  ( x 
C_  A  ->  x  e.  ~P A )
43adantr 466 . . . . 5  |-  ( ( x  C_  A  /\  ps )  ->  x  e. 
~P A )
51, 4syl6bi 231 . . . 4  |-  ( ph  ->  ( x  e.  F  ->  x  e.  ~P A
) )
65ssrdv 3476 . . 3  |-  ( ph  ->  F  C_  ~P A
)
7 isfild.4 . . . 4  |-  ( ph  ->  -.  [. (/)  /  x ]. ps )
8 isfild.2 . . . . . 6  |-  ( ph  ->  A  e.  _V )
91, 8isfildlem 20803 . . . . 5  |-  ( ph  ->  ( (/)  e.  F  <->  (
(/)  C_  A  /\  [. (/)  /  x ]. ps ) ) )
10 simpr 462 . . . . 5  |-  ( (
(/)  C_  A  /\  [. (/)  /  x ]. ps )  ->  [. (/)  /  x ]. ps )
119, 10syl6bi 231 . . . 4  |-  ( ph  ->  ( (/)  e.  F  ->  [. (/)  /  x ]. ps ) )
127, 11mtod 180 . . 3  |-  ( ph  ->  -.  (/)  e.  F )
13 isfild.3 . . . . 5  |-  ( ph  ->  [. A  /  x ]. ps )
14 ssid 3489 . . . . 5  |-  A  C_  A
1513, 14jctil 539 . . . 4  |-  ( ph  ->  ( A  C_  A  /\  [. A  /  x ]. ps ) )
161, 8isfildlem 20803 . . . 4  |-  ( ph  ->  ( A  e.  F  <->  ( A  C_  A  /\  [. A  /  x ]. ps ) ) )
1715, 16mpbird 235 . . 3  |-  ( ph  ->  A  e.  F )
186, 12, 173jca 1185 . 2  |-  ( ph  ->  ( F  C_  ~P A  /\  -.  (/)  e.  F  /\  A  e.  F
) )
19 elpwi 3994 . . . 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 1006 . . . . . . . . . . 11  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  y  C_  A )
2220, 21jctild 545 . . . . . . . . . 10  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( [. z  /  x ]. ps  ->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2322adantld 468 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( (
z  C_  A  /\  [. z  /  x ]. ps )  ->  ( y 
C_  A  /\  [. y  /  x ]. ps )
) )
241, 8isfildlem 20803 . . . . . . . . . 10  |-  ( ph  ->  ( z  e.  F  <->  ( z  C_  A  /\  [. z  /  x ]. ps ) ) )
25243ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( z  e.  F  <->  ( z  C_  A  /\  [. z  /  x ]. ps ) ) )
261, 8isfildlem 20803 . . . . . . . . . 10  |-  ( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
27263ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2823, 25, 273imtr4d 271 . . . . . . . 8  |-  ( (
ph  /\  y  C_  A  /\  z  C_  y
)  ->  ( z  e.  F  ->  y  e.  F ) )
29283expa 1205 . . . . . . 7  |-  ( ( ( ph  /\  y  C_  A )  /\  z  C_  y )  ->  (
z  e.  F  -> 
y  e.  F ) )
3029impancom 441 . . . . . 6  |-  ( ( ( ph  /\  y  C_  A )  /\  z  e.  F )  ->  (
z  C_  y  ->  y  e.  F ) )
3130rexlimdva 2924 . . . . 5  |-  ( (
ph  /\  y  C_  A )  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F ) )
3231ex 435 . . . 4  |-  ( ph  ->  ( y  C_  A  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F ) ) )
3319, 32syl5 33 . . 3  |-  ( ph  ->  ( y  e.  ~P A  ->  ( E. z  e.  F  z  C_  y  ->  y  e.  F
) ) )
3433ralrimiv 2844 . 2  |-  ( ph  ->  A. y  e.  ~P  A ( E. z  e.  F  z  C_  y  ->  y  e.  F
) )
35 ssinss1 3696 . . . . . . 7  |-  ( y 
C_  A  ->  (
y  i^i  z )  C_  A )
3635ad2antrr 730 . . . . . 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 831 . . . . . 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 1206 . . . . . . 7  |-  ( (
ph  /\  ( y  C_  A  /\  z  C_  A ) )  -> 
( ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )  ->  [. (
y  i^i  z )  /  x ]. ps )
)
4140expimpd 606 . . . . . 6  |-  ( ph  ->  ( ( ( y 
C_  A  /\  z  C_  A )  /\  ( [. y  /  x ]. ps  /\  [. z  /  x ]. ps )
)  ->  [. ( y  i^i  z )  /  x ]. ps ) )
4238, 41syl5bi 220 . . . . 5  |-  ( ph  ->  ( ( ( y 
C_  A  /\  [. y  /  x ]. ps )  /\  ( z  C_  A  /\  [. z  /  x ]. ps ) )  ->  [. ( y  i^i  z
)  /  x ]. ps ) )
4337, 42jcad 535 . . . 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 715 . . . 4  |-  ( ph  ->  ( ( y  e.  F  /\  z  e.  F )  <->  ( (
y  C_  A  /\  [. y  /  x ]. ps )  /\  (
z  C_  A  /\  [. z  /  x ]. ps ) ) ) )
451, 8isfildlem 20803 . . . 4  |-  ( ph  ->  ( ( y  i^i  z )  e.  F  <->  ( ( y  i^i  z
)  C_  A  /\  [. ( y  i^i  z
)  /  x ]. ps ) ) )
4643, 44, 453imtr4d 271 . . 3  |-  ( ph  ->  ( ( y  e.  F  /\  z  e.  F )  ->  (
y  i^i  z )  e.  F ) )
4746ralrimivv 2852 . 2  |-  ( ph  ->  A. y  e.  F  A. z  e.  F  ( y  i^i  z
)  e.  F )
48 isfil2 20802 . 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 1189 1  |-  ( ph  ->  F  e.  ( Fil `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087   [.wsbc 3305    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   ` cfv 5601   Filcfil 20791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-fbas 18902  df-fil 20792
This theorem is referenced by:  snfil  20810  fgcl  20824  filuni  20831  cfinfil  20839  csdfil  20840  supfil  20841  fin1aufil  20878
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