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Theorem fsubbas 17562
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 17525 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  =/=  (/) )
2 fvprc 5519 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( fi `  A )  =  (/) )
32necon1ai 2488 . . . . . 6  |-  ( ( fi `  A )  =/=  (/)  ->  A  e.  _V )
41, 3syl 15 . . . . 5  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  e.  _V )
5 ssfii 7172 . . . . 5  |-  ( A  e.  _V  ->  A  C_  ( fi `  A
) )
64, 5syl 15 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ( fi `  A ) )
7 fbsspw 17527 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( fi `  A )  C_  ~P X )
86, 7sstrd 3189 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  A  C_  ~P X )
9 fieq0 7174 . . . . . 6  |-  ( A  e.  _V  ->  ( A  =  (/)  <->  ( fi `  A )  =  (/) ) )
109necon3bid 2481 . . . . 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 17526 . . 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 7182 . . . . 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 4194 . . . . . . . 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 4160 . . . . . . 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 2449 . . . . . 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 7175 . . . . . . . 8  |-  ( ( x  e.  ( fi
`  A )  /\  y  e.  ( fi `  A ) )  -> 
( x  i^i  y
)  e.  ( fi
`  A ) )
29 ssid 3197 . . . . . . . 8  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
30 sseq1 3199 . . . . . . . . 9  |-  ( z  =  ( x  i^i  y )  ->  (
z  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
3130rspcev 2884 . . . . . . . 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 2609 . . . . . 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 17530 . . . . 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 1684    =/= wne 2446    e/ wnel 2447   A.wral 2543   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   ` cfv 5255   ficfi 7164   fBascfbas 17518
This theorem is referenced by:  isufil2  17603  ufileu  17614  filufint  17615  fmfnfm  17653  hausflim  17676  flimclslem  17679  fclsfnflim  17722  flimfnfcls  17723  fclscmp  17725  efilcp  25552  fgsb2  25555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-fbas 17520
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