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Theorem fsubbas 17901
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 17864 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  =/=  (/) )
2 fvprc 5724 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( fi `  A )  =  (/) )
32necon1ai 2648 . . . . . 6  |-  ( ( fi `  A )  =/=  (/)  ->  A  e.  _V )
41, 3syl 16 . . . . 5  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  e.  _V )
5 ssfii 7426 . . . . 5  |-  ( A  e.  _V  ->  A  C_  ( fi `  A
) )
64, 5syl 16 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ( fi `  A ) )
7 fbsspw 17866 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  C_  ~P X )
86, 7sstrd 3360 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ~P X )
9 fieq0 7428 . . . . . 6  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  ( fi `  A )  =  (/) ) )
109necon3bid 2638 . . . . 5  |-  ( A  e.  _V  ->  ( A  =/=  (/)  <->  ( fi `  A )  =/=  (/) ) )
1110biimpar 473 . . . 4  |-  ( ( A  e.  _V  /\  ( fi `  A )  =/=  (/) )  ->  A  =/=  (/) )
124, 1, 11syl2anc 644 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  =/=  (/) )
13 0nelfb 17865 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  -.  (/)  e.  ( fi `  A ) )
148, 12, 133jca 1135 . 2  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) )
15 simpr1 964 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ~P X )
16 fipwss 7436 . . . . 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 4385 . . . . . . . 8  |-  ( X  e.  V  ->  ~P X  e.  _V )
1918adantr 453 . . . . . . 7  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ~P X  e. 
_V )
2019, 15ssexd 4352 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  e.  _V )
21 simpr2 965 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  =/=  (/) )
2210biimpa 472 . . . . . 6  |-  ( ( A  e.  _V  /\  A  =/=  (/) )  ->  ( fi `  A )  =/=  (/) )
2320, 21, 22syl2anc 644 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  =/=  (/) )
24 simpr3 966 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  -.  (/)  e.  ( fi `  A ) )
25 df-nel 2604 . . . . . 6  |-  ( (/)  e/  ( fi `  A
)  <->  -.  (/)  e.  ( fi `  A ) )
2624, 25sylibr 205 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  (/)  e/  ( fi
`  A ) )
27 fiin 7429 . . . . . . . 8  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  -> 
( x  i^i  y
)  e.  ( fi
`  A ) )
28 ssid 3369 . . . . . . . 8  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
29 sseq1 3371 . . . . . . . . 9  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
3029rspcev 3054 . . . . . . . 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 645 . . . . . . 7  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  ->  E. z  e.  ( fi `  A ) z 
C_  ( x  i^i  y ) )
3231rgen2a 2774 . . . . . 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 1135 . . . 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 17869 . . . . 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 453 . . . 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 890 . . 3  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  e.  (
fBas `  X )
)
3837ex 425 . 2  |-  ( X  e.  V  ->  (
( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) )  -> 
( fi `  A
)  e.  ( fBas `  X ) ) )
3914, 38impbid2 197 1  |-  ( X  e.  V  ->  (
( fi `  A
)  e.  ( fBas `  X )  <->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726    =/= wne 2601    e/ wnel 2602   A.wral 2707   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   ` cfv 5456   ficfi 7417   fBascfbas 16691
This theorem is referenced by:  isufil2  17942  ufileu  17953  filufint  17954  fmfnfm  17992  hausflim  18015  flimclslem  18018  fclsfnflim  18061  flimfnfcls  18062  fclscmp  18064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-fin 7115  df-fi 7418  df-fbas 16701
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