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Theorem filin 20879
Description: A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filin  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  e.  F )

Proof of Theorem filin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 filfbas 20873 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbasssin 20861 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
31, 2syl3an1 1304 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
4 inss1 3619 . . . . 5  |-  ( A  i^i  B )  C_  A
5 filelss 20877 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
64, 5syl5ss 3410 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( A  i^i  B )  C_  X )
7 filss 20878 . . . . . . . 8  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  ( A  i^i  B ) 
C_  X  /\  x  C_  ( A  i^i  B
) ) )  -> 
( A  i^i  B
)  e.  F )
873exp2 1230 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( ( A  i^i  B ) 
C_  X  ->  (
x  C_  ( A  i^i  B )  ->  ( A  i^i  B )  e.  F ) ) ) )
98com23 81 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( ( A  i^i  B )  C_  X  ->  ( x  e.  F  ->  ( x  C_  ( A  i^i  B
)  ->  ( A  i^i  B )  e.  F
) ) ) )
109imp 435 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  i^i  B )  C_  X )  ->  (
x  e.  F  -> 
( x  C_  ( A  i^i  B )  -> 
( A  i^i  B
)  e.  F ) ) )
1110rexlimdv 2850 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  ( A  i^i  B )  C_  X )  ->  ( E. x  e.  F  x  C_  ( A  i^i  B )  ->  ( A  i^i  B )  e.  F
) )
126, 11syldan 477 . . 3  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( E. x  e.  F  x  C_  ( A  i^i  B )  ->  ( A  i^i  B )  e.  F
) )
13123adant3 1029 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( E. x  e.  F  x  C_  ( A  i^i  B )  ->  ( A  i^i  B )  e.  F
) )
143, 13mpd 15 1  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  e.  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 986    e. wcel 1890   E.wrex 2737    i^i cin 3370    C_ wss 3371   ` cfv 5560   fBascfbas 18968   Filcfil 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fv 5568  df-fbas 18977  df-fil 20871
This theorem is referenced by:  isfil2  20881  filfi  20884  filinn0  20885  infil  20888  filcon  20908  filuni  20910  trfil2  20912  trfilss  20914  ufprim  20934  filufint  20945  rnelfmlem  20977  rnelfm  20978  fmfnfmlem2  20980  fmfnfmlem3  20981  fmfnfmlem4  20982  fmfnfm  20983  txflf  21031  fclsrest  21049  metust  21583  filnetlem3  31041
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