Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fgsb2 Unicode version

Theorem fgsb2 25555
Description: Filter generated by a subbasis  A. Bourbaki TG I.37 paragraph above prop. 1. (The theorem has been slightly modified because the definitions of the empty set are different in Bourbaki and Metamath.) (Contributed by FL, 27-Apr-2008.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
fgsb2  |-  ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  { x  e. 
~P X  |  E. y  e.  ( fi `  A ) y  C_  x }  e.  ( Fil `  X ) ) )
Distinct variable groups:    x, y, A    x, X, y

Proof of Theorem fgsb2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . . 4  |-  ( ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  /\  -.  (/)  e.  ( fi
`  A ) )  ->  A  C_  ~P X )
2 simpl3 960 . . . 4  |-  ( ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  /\  -.  (/)  e.  ( fi
`  A ) )  ->  A  =/=  (/) )
3 simpr 447 . . . 4  |-  ( ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  /\  -.  (/)  e.  ( fi
`  A ) )  ->  -.  (/)  e.  ( fi `  A ) )
4 simpl2 959 . . . . 5  |-  ( ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  /\  -.  (/)  e.  ( fi
`  A ) )  ->  X  e.  _V )
5 fsubbas 17562 . . . . 5  |-  ( X  e.  _V  ->  (
( fi `  A
)  e.  ( fBas `  X )  <->  ( A  C_ 
~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )
64, 5syl 15 . . . 4  |-  ( ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  /\  -.  (/)  e.  ( fi
`  A ) )  ->  ( ( fi
`  A )  e.  ( fBas `  X
)  <->  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi `  A ) ) ) )
71, 2, 3, 6mpbir3and 1135 . . 3  |-  ( ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  /\  -.  (/)  e.  ( fi
`  A ) )  ->  ( fi `  A )  e.  (
fBas `  X )
)
87ex 423 . 2  |-  ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  ( fi `  A )  e.  (
fBas `  X )
) )
9 sseq2 3200 . . . . . . 7  |-  ( x  =  z  ->  (
y  C_  x  <->  y  C_  z ) )
109rexbidv 2564 . . . . . 6  |-  ( x  =  z  ->  ( E. y  e.  ( fi `  A ) y 
C_  x  <->  E. y  e.  ( fi `  A
) y  C_  z
) )
1110elrab 2923 . . . . 5  |-  ( z  e.  { x  e. 
~P X  |  E. y  e.  ( fi `  A ) y  C_  x }  <->  ( z  e. 
~P X  /\  E. y  e.  ( fi `  A ) y  C_  z ) )
12 elfg 17566 . . . . . 6  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( z  e.  ( X filGen ( fi
`  A ) )  <-> 
( z  C_  X  /\  E. y  e.  ( fi `  A ) y  C_  z )
) )
13 vex 2791 . . . . . . . 8  |-  z  e. 
_V
1413elpw 3631 . . . . . . 7  |-  ( z  e.  ~P X  <->  z  C_  X )
1514anbi1i 676 . . . . . 6  |-  ( ( z  e.  ~P X  /\  E. y  e.  ( fi `  A ) y  C_  z )  <->  ( z  C_  X  /\  E. y  e.  ( fi
`  A ) y 
C_  z ) )
1612, 15syl6rbbr 255 . . . . 5  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( (
z  e.  ~P X  /\  E. y  e.  ( fi `  A ) y  C_  z )  <->  z  e.  ( X filGen ( fi `  A ) ) ) )
1711, 16syl5bb 248 . . . 4  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( z  e.  { x  e.  ~P X  |  E. y  e.  ( fi `  A
) y  C_  x } 
<->  z  e.  ( X
filGen ( fi `  A
) ) ) )
1817eqrdv 2281 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  { x  e.  ~P X  |  E. y  e.  ( fi `  A ) y  C_  x }  =  ( X filGen ( fi `  A ) ) )
19 fgcl 17573 . . 3  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  ( X filGen ( fi `  A
) )  e.  ( Fil `  X ) )
2018, 19eqeltrd 2357 . 2  |-  ( ( fi `  A )  e.  ( fBas `  X
)  ->  { x  e.  ~P X  |  E. y  e.  ( fi `  A ) y  C_  x }  e.  ( Fil `  X ) )
218, 20syl6 29 1  |-  ( ( A  C_  ~P X  /\  X  e.  _V  /\  A  =/=  (/) )  -> 
( -.  (/)  e.  ( fi `  A )  ->  { x  e. 
~P X  |  E. y  e.  ( fi `  A ) y  C_  x }  e.  ( Fil `  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   ` cfv 5255  (class class class)co 5858   ficfi 7164   fBascfbas 17518   filGencfg 17519   Filcfil 17540
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  df-fg 17521  df-fil 17541
  Copyright terms: Public domain W3C validator