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Theorem fbssint 17870
Description: A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
fbssint  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  E. x  e.  F  x  C_  |^| A
)
Distinct variable groups:    x, A    x, F    x, B

Proof of Theorem fbssint
StepHypRef Expression
1 fbasne0 17862 . . . . . 6  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )
2 n0 3637 . . . . . 6  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
31, 2sylib 189 . . . . 5  |-  ( F  e.  ( fBas `  B
)  ->  E. x  x  e.  F )
4 ssv 3368 . . . . . . . 8  |-  x  C_  _V
54jctr 527 . . . . . . 7  |-  ( x  e.  F  ->  (
x  e.  F  /\  x  C_  _V ) )
65eximi 1585 . . . . . 6  |-  ( E. x  x  e.  F  ->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
7 df-rex 2711 . . . . . 6  |-  ( E. x  e.  F  x 
C_  _V  <->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
86, 7sylibr 204 . . . . 5  |-  ( E. x  x  e.  F  ->  E. x  e.  F  x  C_  _V )
93, 8syl 16 . . . 4  |-  ( F  e.  ( fBas `  B
)  ->  E. x  e.  F  x  C_  _V )
10 inteq 4053 . . . . . . 7  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
11 int0 4064 . . . . . . 7  |-  |^| (/)  =  _V
1210, 11syl6eq 2484 . . . . . 6  |-  ( A  =  (/)  ->  |^| A  =  _V )
1312sseq2d 3376 . . . . 5  |-  ( A  =  (/)  ->  ( x 
C_  |^| A  <->  x  C_  _V ) )
1413rexbidv 2726 . . . 4  |-  ( A  =  (/)  ->  ( E. x  e.  F  x 
C_  |^| A  <->  E. x  e.  F  x  C_  _V ) )
159, 14syl5ibrcom 214 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
16153ad2ant1 978 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
17 simpl1 960 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  F  e.  ( fBas `  B ) )
18 simpl2 961 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  C_  F )
19 simpr 448 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
20 simpl3 962 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
21 elfir 7420 . . . . 5  |-  ( ( F  e.  ( fBas `  B )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  F ) )
2217, 18, 19, 20, 21syl13anc 1186 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  |^| A  e.  ( fi
`  F ) )
23 fbssfi 17869 . . . 4  |-  ( ( F  e.  ( fBas `  B )  /\  |^| A  e.  ( fi `  F ) )  ->  E. x  e.  F  x  C_  |^| A )
2417, 22, 23syl2anc 643 . . 3  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  F  x  C_  |^| A )
2524ex 424 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =/=  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
2616, 25pm2.61dne 2681 1  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  E. x  e.  F  x  C_  |^| A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   _Vcvv 2956    C_ wss 3320   (/)c0 3628   |^|cint 4050   ` cfv 5454   Fincfn 7109   ficfi 7415   fBascfbas 16689
This theorem is referenced by:  fbasfip  17900
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-fin 7113  df-fi 7416  df-fbas 16699
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