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Theorem islinds 27382
Description: Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
islinds.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
islinds  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF 
W ) ) )

Proof of Theorem islinds
Dummy variables  s  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2809 . . . . 5  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5541 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32pweqd 3643 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
4 breq2 4043 . . . . . . 7  |-  ( w  =  W  ->  (
(  _I  |`  s
) LIndF  w  <->  (  _I  |`  s
) LIndF  W ) )
53, 4rabeqbidv 2796 . . . . . 6  |-  ( w  =  W  ->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF  w }  =  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } )
6 df-linds 27380 . . . . . 6  |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s
) LIndF  w } )
7 fvex 5555 . . . . . . . 8  |-  ( Base `  W )  e.  _V
87pwex 4209 . . . . . . 7  |-  ~P ( Base `  W )  e. 
_V
98rabex 4181 . . . . . 6  |-  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W }  e.  _V
105, 6, 9fvmpt 5618 . . . . 5  |-  ( W  e.  _V  ->  (LIndS `  W )  =  {
s  e.  ~P ( Base `  W )  |  (  _I  |`  s
) LIndF  W } )
111, 10syl 15 . . . 4  |-  ( W  e.  V  ->  (LIndS `  W )  =  {
s  e.  ~P ( Base `  W )  |  (  _I  |`  s
) LIndF  W } )
1211eleq2d 2363 . . 3  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  X  e.  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } ) )
13 reseq2 4966 . . . . 5  |-  ( s  =  X  ->  (  _I  |`  s )  =  (  _I  |`  X ) )
1413breq1d 4049 . . . 4  |-  ( s  =  X  ->  (
(  _I  |`  s
) LIndF  W  <->  (  _I  |`  X ) LIndF 
W ) )
1514elrab 2936 . . 3  |-  ( X  e.  { s  e. 
~P ( Base `  W
)  |  (  _I  |`  s ) LIndF  W }  <->  ( X  e.  ~P ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W ) )
1612, 15syl6bb 252 . 2  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  ( X  e. 
~P ( Base `  W
)  /\  (  _I  |`  X ) LIndF  W ) ) )
177elpw2 4191 . . . 4  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  ( Base `  W ) )
18 islinds.b . . . . 5  |-  B  =  ( Base `  W
)
1918sseq2i 3216 . . . 4  |-  ( X 
C_  B  <->  X  C_  ( Base `  W ) )
2017, 19bitr4i 243 . . 3  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  B
)
2120anbi1i 676 . 2  |-  ( ( X  e.  ~P ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W )  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF  W ) )
2216, 21syl6bb 252 1  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF 
W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   LIndF clindf 27377  LIndSclinds 27378
This theorem is referenced by:  linds1  27383  linds2  27384  islinds2  27386  lindsss  27397  lindsmm  27401  lsslinds  27404
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-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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-linds 27380
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