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Theorem fsubbas 19440
Description: A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fsubbas  |-  ( X  e.  V  ->  (
( fi `  A
)  e.  ( fBas `  X )  <->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )

Proof of Theorem fsubbas
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fbasne0 19403 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  =/=  (/) )
2 fvprc 5685 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( fi `  A )  =  (/) )
32necon1ai 2653 . . . . . 6  |-  ( ( fi `  A )  =/=  (/)  ->  A  e.  _V )
41, 3syl 16 . . . . 5  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  e.  _V )
5 ssfii 7669 . . . . 5  |-  ( A  e.  _V  ->  A  C_  ( fi `  A
) )
64, 5syl 16 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ( fi `  A ) )
7 fbsspw 19405 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  C_  ~P X )
86, 7sstrd 3366 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ~P X )
9 fieq0 7671 . . . . . 6  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  ( fi `  A )  =  (/) ) )
109necon3bid 2643 . . . . 5  |-  ( A  e.  _V  ->  ( A  =/=  (/)  <->  ( fi `  A )  =/=  (/) ) )
1110biimpar 485 . . . 4  |-  ( ( A  e.  _V  /\  ( fi `  A )  =/=  (/) )  ->  A  =/=  (/) )
124, 1, 11syl2anc 661 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  =/=  (/) )
13 0nelfb 19404 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  -.  (/)  e.  ( fi `  A ) )
148, 12, 133jca 1168 . 2  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) )
15 simpr1 994 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ~P X )
16 fipwss 7679 . . . . 5  |-  ( A 
C_  ~P X  ->  ( fi `  A )  C_  ~P X )
1715, 16syl 16 . . . 4  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  C_  ~P X )
18 pwexg 4476 . . . . . . . 8  |-  ( X  e.  V  ->  ~P X  e.  _V )
1918adantr 465 . . . . . . 7  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ~P X  e. 
_V )
2019, 15ssexd 4439 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  e.  _V )
21 simpr2 995 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  =/=  (/) )
2210biimpa 484 . . . . . 6  |-  ( ( A  e.  _V  /\  A  =/=  (/) )  ->  ( fi `  A )  =/=  (/) )
2320, 21, 22syl2anc 661 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  =/=  (/) )
24 simpr3 996 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  -.  (/)  e.  ( fi `  A ) )
25 df-nel 2609 . . . . . 6  |-  ( (/)  e/  ( fi `  A
)  <->  -.  (/)  e.  ( fi `  A ) )
2624, 25sylibr 212 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  (/)  e/  ( fi
`  A ) )
27 fiin 7672 . . . . . . . 8  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  -> 
( x  i^i  y
)  e.  ( fi
`  A ) )
28 ssid 3375 . . . . . . . 8  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
29 sseq1 3377 . . . . . . . . 9  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
3029rspcev 3073 . . . . . . . 8  |-  ( ( ( x  i^i  y
)  e.  ( fi
`  A )  /\  ( x  i^i  y
)  C_  ( x  i^i  y ) )  ->  E. z  e.  ( fi `  A ) z 
C_  ( x  i^i  y ) )
3127, 28, 30sylancl 662 . . . . . . 7  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  ->  E. z  e.  ( fi `  A ) z 
C_  ( x  i^i  y ) )
3231rgen2a 2782 . . . . . 6  |-  A. x  e.  ( fi `  A
) A. y  e.  ( fi `  A
) E. z  e.  ( fi `  A
) z  C_  (
x  i^i  y )
3332a1i 11 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A. x  e.  ( fi `  A ) A. y  e.  ( fi `  A ) E. z  e.  ( fi `  A ) z  C_  ( x  i^i  y ) )
3423, 26, 333jca 1168 . . . 4  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( ( fi
`  A )  =/=  (/)  /\  (/)  e/  ( fi
`  A )  /\  A. x  e.  ( fi
`  A ) A. y  e.  ( fi `  A ) E. z  e.  ( fi `  A
) z  C_  (
x  i^i  y )
) )
35 isfbas2 19408 . . . . 5  |-  ( X  e.  V  ->  (
( fi `  A
)  e.  ( fBas `  X )  <->  ( ( fi `  A )  C_  ~P X  /\  (
( fi `  A
)  =/=  (/)  /\  (/)  e/  ( fi `  A )  /\  A. x  e.  ( fi
`  A ) A. y  e.  ( fi `  A ) E. z  e.  ( fi `  A
) z  C_  (
x  i^i  y )
) ) ) )
3635adantr 465 . . . 4  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( ( fi
`  A )  e.  ( fBas `  X
)  <->  ( ( fi
`  A )  C_  ~P X  /\  (
( fi `  A
)  =/=  (/)  /\  (/)  e/  ( fi `  A )  /\  A. x  e.  ( fi
`  A ) A. y  e.  ( fi `  A ) E. z  e.  ( fi `  A
) z  C_  (
x  i^i  y )
) ) ) )
3717, 34, 36mpbir2and 913 . . 3  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  e.  (
fBas `  X )
)
3837ex 434 . 2  |-  ( X  e.  V  ->  (
( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) )  -> 
( fi `  A
)  e.  ( fBas `  X ) ) )
3914, 38impbid2 204 1  |-  ( X  e.  V  ->  (
( fi `  A
)  e.  ( fBas `  X )  <->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1756    =/= wne 2606    e/ wnel 2607   A.wral 2715   E.wrex 2716   _Vcvv 2972    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   ` cfv 5418   ficfi 7660   fBascfbas 17804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-fin 7314  df-fi 7661  df-fbas 17814
This theorem is referenced by:  isufil2  19481  ufileu  19492  filufint  19493  fmfnfm  19531  hausflim  19554  flimclslem  19557  fclsfnflim  19600  flimfnfcls  19601  fclscmp  19603
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