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Theorem filin 17565
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 17559 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbasssin 17547 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
31, 2syl3an1 1215 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
4 inss1 3402 . . . . 5  |-  ( A  i^i  B )  C_  A
5 filelss 17563 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
64, 5syl5ss 3203 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( A  i^i  B )  C_  X )
7 filss 17564 . . . . . . . 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 1169 . . . . . . 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 72 . . . . . 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 418 . . . . 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 2679 . . . 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 456 . . 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 975 . 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 14 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 358    /\ w3a 934    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   ` cfv 5271   fBascfbas 17534   Filcfil 17556
This theorem is referenced by:  isfil2  17567  filfi  17570  filinn0  17571  infil  17574  filcon  17594  filuni  17596  trfil2  17598  trfilss  17600  ufprim  17620  filufint  17631  rnelfmlem  17663  rnelfm  17664  fmfnfmlem2  17666  fmfnfmlem3  17667  fmfnfmlem4  17668  fmfnfm  17669  txflf  17717  fclsrest  17735  limptlimpr2lem2  25678  lvsovso  25729  filnetlem3  26432
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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