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Theorem filin 17878
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 17872 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbasssin 17860 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
31, 2syl3an1 1217 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
4 inss1 3553 . . . . 5  |-  ( A  i^i  B )  C_  A
5 filelss 17876 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
64, 5syl5ss 3351 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( A  i^i  B )  C_  X )
7 filss 17877 . . . . . . . 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 1171 . . . . . . 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 74 . . . . . 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 419 . . . . 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 2821 . . . 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 457 . . 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 977 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   E.wrex 2698    i^i cin 3311    C_ wss 3312   ` cfv 5446   fBascfbas 16681   Filcfil 17869
This theorem is referenced by:  isfil2  17880  filfi  17883  filinn0  17884  infil  17887  filcon  17907  filuni  17909  trfil2  17911  trfilss  17913  ufprim  17933  filufint  17944  rnelfmlem  17976  rnelfm  17977  fmfnfmlem2  17979  fmfnfmlem3  17980  fmfnfmlem4  17981  fmfnfm  17982  txflf  18030  fclsrest  18048  metustOLD  18589  metust  18590  filnetlem3  26400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-fbas 16691  df-fil 17870
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