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Theorem filin 20225
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 20219 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbasssin 20207 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
31, 2syl3an1 1260 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
4 inss1 3701 . . . . 5  |-  ( A  i^i  B )  C_  A
5 filelss 20223 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
64, 5syl5ss 3498 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( A  i^i  B )  C_  X )
7 filss 20224 . . . . . . . 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 1213 . . . . . . 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 78 . . . . . 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 429 . . . . 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 2931 . . . 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 470 . . 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 1015 . 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 369    /\ w3a 972    e. wcel 1802   E.wrex 2792    i^i cin 3458    C_ wss 3459   ` cfv 5575   fBascfbas 18277   Filcfil 20216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fv 5583  df-fbas 18287  df-fil 20217
This theorem is referenced by:  isfil2  20227  filfi  20230  filinn0  20231  infil  20234  filcon  20254  filuni  20256  trfil2  20258  trfilss  20260  ufprim  20280  filufint  20291  rnelfmlem  20323  rnelfm  20324  fmfnfmlem2  20326  fmfnfmlem3  20327  fmfnfmlem4  20328  fmfnfm  20329  txflf  20377  fclsrest  20395  metustOLD  20940  metust  20941  filnetlem3  30170
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