MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fbssint Unicode version

Theorem fbssint 17585
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 17577 . . . . . 6  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )
2 n0 3498 . . . . . 6  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
31, 2sylib 188 . . . . 5  |-  ( F  e.  ( fBas `  B
)  ->  E. x  x  e.  F )
4 ssv 3232 . . . . . . . 8  |-  x  C_  _V
54jctr 526 . . . . . . 7  |-  ( x  e.  F  ->  (
x  e.  F  /\  x  C_  _V ) )
65eximi 1567 . . . . . 6  |-  ( E. x  x  e.  F  ->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
7 df-rex 2583 . . . . . 6  |-  ( E. x  e.  F  x 
C_  _V  <->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
86, 7sylibr 203 . . . . 5  |-  ( E. x  x  e.  F  ->  E. x  e.  F  x  C_  _V )
93, 8syl 15 . . . 4  |-  ( F  e.  ( fBas `  B
)  ->  E. x  e.  F  x  C_  _V )
10 inteq 3902 . . . . . . 7  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
11 int0 3913 . . . . . . 7  |-  |^| (/)  =  _V
1210, 11syl6eq 2364 . . . . . 6  |-  ( A  =  (/)  ->  |^| A  =  _V )
1312sseq2d 3240 . . . . 5  |-  ( A  =  (/)  ->  ( x 
C_  |^| A  <->  x  C_  _V ) )
1413rexbidv 2598 . . . 4  |-  ( A  =  (/)  ->  ( E. x  e.  F  x 
C_  |^| A  <->  E. x  e.  F  x  C_  _V ) )
159, 14syl5ibrcom 213 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
16153ad2ant1 976 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
17 simpl1 958 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  F  e.  ( fBas `  B ) )
18 simpl2 959 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  C_  F )
19 simpr 447 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
20 simpl3 960 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
21 elfir 7214 . . . . 5  |-  ( ( F  e.  ( fBas `  B )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  F ) )
2217, 18, 19, 20, 21syl13anc 1184 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  |^| A  e.  ( fi
`  F ) )
23 fbssfi 17584 . . . 4  |-  ( ( F  e.  ( fBas `  B )  /\  |^| A  e.  ( fi `  F ) )  ->  E. x  e.  F  x  C_  |^| A )
2417, 22, 23syl2anc 642 . . 3  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  F  x  C_  |^| A )
2524ex 423 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =/=  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
2616, 25pm2.61dne 2556 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 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   _Vcvv 2822    C_ wss 3186   (/)c0 3489   |^|cint 3899   ` cfv 5292   Fincfn 6906   ficfi 7209   fBascfbas 16421
This theorem is referenced by:  fbasfip  17615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-fin 6910  df-fi 7210  df-fbas 16429
  Copyright terms: Public domain W3C validator