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Theorem fsubbas 20195
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 20158 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  =/=  (/) )
2 fvprc 5860 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( fi `  A )  =  (/) )
32necon1ai 2698 . . . . . 6  |-  ( ( fi `  A )  =/=  (/)  ->  A  e.  _V )
41, 3syl 16 . . . . 5  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  e.  _V )
5 ssfii 7880 . . . . 5  |-  ( A  e.  _V  ->  A  C_  ( fi `  A
) )
64, 5syl 16 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ( fi `  A ) )
7 fbsspw 20160 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  C_  ~P X )
86, 7sstrd 3514 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ~P X )
9 fieq0 7882 . . . . . 6  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  ( fi `  A )  =  (/) ) )
109necon3bid 2725 . . . . 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 20159 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  -.  (/)  e.  ( fi `  A ) )
148, 12, 133jca 1176 . 2  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) )
15 simpr1 1002 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ~P X )
16 fipwss 7890 . . . . 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 4631 . . . . . . . 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 4594 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  e.  _V )
21 simpr2 1003 . . . . . 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 1004 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  -.  (/)  e.  ( fi `  A ) )
25 df-nel 2665 . . . . . 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 7883 . . . . . . . 8  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  -> 
( x  i^i  y
)  e.  ( fi
`  A ) )
28 ssid 3523 . . . . . . . 8  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
29 sseq1 3525 . . . . . . . . 9  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
3029rspcev 3214 . . . . . . . 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 2891 . . . . . 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 1176 . . . 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 20163 . . . . 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 920 . . 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 973    e. wcel 1767    =/= wne 2662    e/ wnel 2663   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   ` cfv 5588   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:  isufil2  20236  ufileu  20247  filufint  20248  fmfnfm  20286  hausflim  20309  flimclslem  20312  fclsfnflim  20355  flimfnfcls  20356  fclscmp  20358
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