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Theorem fbssint 20166
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 20158 . . . . . 6  |-  ( F  e.  ( fBas `  B
)  ->  F  =/=  (/) )
2 n0 3794 . . . . . 6  |-  ( F  =/=  (/)  <->  E. x  x  e.  F )
31, 2sylib 196 . . . . 5  |-  ( F  e.  ( fBas `  B
)  ->  E. x  x  e.  F )
4 ssv 3524 . . . . . . . 8  |-  x  C_  _V
54jctr 542 . . . . . . 7  |-  ( x  e.  F  ->  (
x  e.  F  /\  x  C_  _V ) )
65eximi 1635 . . . . . 6  |-  ( E. x  x  e.  F  ->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
7 df-rex 2820 . . . . . 6  |-  ( E. x  e.  F  x 
C_  _V  <->  E. x ( x  e.  F  /\  x  C_ 
_V ) )
86, 7sylibr 212 . . . . 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 4285 . . . . . . 7  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
11 int0 4296 . . . . . . 7  |-  |^| (/)  =  _V
1210, 11syl6eq 2524 . . . . . 6  |-  ( A  =  (/)  ->  |^| A  =  _V )
1312sseq2d 3532 . . . . 5  |-  ( A  =  (/)  ->  ( x 
C_  |^| A  <->  x  C_  _V ) )
1413rexbidv 2973 . . . 4  |-  ( A  =  (/)  ->  ( E. x  e.  F  x 
C_  |^| A  <->  E. x  e.  F  x  C_  _V ) )
159, 14syl5ibrcom 222 . . 3  |-  ( F  e.  ( fBas `  B
)  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
16153ad2ant1 1017 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
17 simpl1 999 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  F  e.  ( fBas `  B ) )
18 simpl2 1000 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  C_  F )
19 simpr 461 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  =/=  (/) )
20 simpl3 1001 . . . . 5  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  A  e.  Fin )
21 elfir 7876 . . . . 5  |-  ( ( F  e.  ( fBas `  B )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e. 
Fin ) )  ->  |^| A  e.  ( fi
`  F ) )
2217, 18, 19, 20, 21syl13anc 1230 . . . 4  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  |^| A  e.  ( fi
`  F ) )
23 fbssfi 20165 . . . 4  |-  ( ( F  e.  ( fBas `  B )  /\  |^| A  e.  ( fi `  F ) )  ->  E. x  e.  F  x  C_  |^| A )
2417, 22, 23syl2anc 661 . . 3  |-  ( ( ( F  e.  (
fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  /\  A  =/=  (/) )  ->  E. x  e.  F  x  C_  |^| A )
2524ex 434 . 2  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  ( A  =/=  (/)  ->  E. x  e.  F  x  C_  |^| A
) )
2616, 25pm2.61dne 2784 1  |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e. 
Fin )  ->  E. x  e.  F  x  C_  |^| A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   |^|cint 4282   ` cfv 5588   Fincfn 7517   ficfi 7871   fBascfbas 18217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-fin 7521  df-fi 7872  df-fbas 18227
This theorem is referenced by:  fbasfip  20196
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