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Theorem fsubbas 17578
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 17541 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  =/=  (/) )
2 fvprc 5535 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( fi `  A )  =  (/) )
32necon1ai 2501 . . . . . 6  |-  ( ( fi `  A )  =/=  (/)  ->  A  e.  _V )
41, 3syl 15 . . . . 5  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  e.  _V )
5 ssfii 7188 . . . . 5  |-  ( A  e.  _V  ->  A  C_  ( fi `  A
) )
64, 5syl 15 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ( fi `  A ) )
7 fbsspw 17543 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  C_  ~P X )
86, 7sstrd 3202 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ~P X )
9 fieq0 7190 . . . . . 6  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  ( fi `  A )  =  (/) ) )
109necon3bid 2494 . . . . 5  |-  ( A  e.  _V  ->  ( A  =/=  (/)  <->  ( fi `  A )  =/=  (/) ) )
1110biimpar 471 . . . 4  |-  ( ( A  e.  _V  /\  ( fi `  A )  =/=  (/) )  ->  A  =/=  (/) )
124, 1, 11syl2anc 642 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  =/=  (/) )
13 0nelfb 17542 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  -.  (/)  e.  ( fi `  A ) )
148, 12, 133jca 1132 . 2  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) )
15 simpr1 961 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  C_  ~P X )
16 fipwss 7198 . . . . 5  |-  ( A 
C_  ~P X  ->  ( fi `  A )  C_  ~P X )
1715, 16syl 15 . . . 4  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  C_  ~P X )
18 pwexg 4210 . . . . . . . 8  |-  ( X  e.  V  ->  ~P X  e.  _V )
1918adantr 451 . . . . . . 7  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ~P X  e. 
_V )
20 ssexg 4176 . . . . . . 7  |-  ( ( A  C_  ~P X  /\  ~P X  e.  _V )  ->  A  e.  _V )
2115, 19, 20syl2anc 642 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  e.  _V )
22 simpr2 962 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  A  =/=  (/) )
2310biimpa 470 . . . . . 6  |-  ( ( A  e.  _V  /\  A  =/=  (/) )  ->  ( fi `  A )  =/=  (/) )
2421, 22, 23syl2anc 642 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  =/=  (/) )
25 simpr3 963 . . . . . 6  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  -.  (/)  e.  ( fi `  A ) )
26 df-nel 2462 . . . . . 6  |-  ( (/)  e/  ( fi `  A
)  <->  -.  (/)  e.  ( fi `  A ) )
2725, 26sylibr 203 . . . . 5  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  (/)  e/  ( fi
`  A ) )
28 fiin 7191 . . . . . . . 8  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  -> 
( x  i^i  y
)  e.  ( fi
`  A ) )
29 ssid 3210 . . . . . . . 8  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
30 sseq1 3212 . . . . . . . . 9  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
3130rspcev 2897 . . . . . . . 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 ) )
3228, 29, 31sylancl 643 . . . . . . 7  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  ->  E. z  e.  ( fi `  A ) z 
C_  ( x  i^i  y ) )
3332rgen2a 2622 . . . . . 6  |-  A. x  e.  ( fi `  A
) A. y  e.  ( fi `  A
) E. z  e.  ( fi `  A
) z  C_  (
x  i^i  y )
3433a1i 10 . . . . 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 ) )
3524, 27, 343jca 1132 . . . 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 )
) )
36 isfbas2 17546 . . . . 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 )
) ) ) )
3736adantr 451 . . . 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 )
) ) ) )
3817, 35, 37mpbir2and 888 . . 3  |-  ( ( X  e.  V  /\  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) ) )  ->  ( fi `  A )  e.  (
fBas `  X )
)
3938ex 423 . 2  |-  ( X  e.  V  ->  (
( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/) 
e.  ( fi `  A ) )  -> 
( fi `  A
)  e.  ( fBas `  X ) ) )
4014, 39impbid2 195 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 176    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459    e/ wnel 2460   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   ` cfv 5271   ficfi 7180   fBascfbas 17534
This theorem is referenced by:  isufil2  17619  ufileu  17630  filufint  17631  fmfnfm  17669  hausflim  17692  flimclslem  17695  fclsfnflim  17738  flimfnfcls  17739  fclscmp  17741  efilcp  25655  fgsb2  25658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-fbas 17536
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