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Theorem filin 20521
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 20515 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 fbasssin 20503 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
31, 2syl3an1 1259 . 2  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
4 inss1 3704 . . . . 5  |-  ( A  i^i  B )  C_  A
5 filelss 20519 . . . . 5  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  A  C_  X )
64, 5syl5ss 3500 . . . 4  |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F )  ->  ( A  i^i  B )  C_  X )
7 filss 20520 . . . . . . . 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 1212 . . . . . . 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 427 . . . . 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 2944 . . . 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 468 . . 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 1014 . 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 367    /\ w3a 971    e. wcel 1823   E.wrex 2805    i^i cin 3460    C_ wss 3461   ` cfv 5570   fBascfbas 18601   Filcfil 20512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-fbas 18611  df-fil 20513
This theorem is referenced by:  isfil2  20523  filfi  20526  filinn0  20527  infil  20530  filcon  20550  filuni  20552  trfil2  20554  trfilss  20556  ufprim  20576  filufint  20587  rnelfmlem  20619  rnelfm  20620  fmfnfmlem2  20622  fmfnfmlem3  20623  fmfnfmlem4  20624  fmfnfm  20625  txflf  20673  fclsrest  20691  metustOLD  21236  metust  21237  filnetlem3  30438
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