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Theorem islinds 27279
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 2796 . . . . 5  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32pweqd 3630 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
4 breq2 4027 . . . . . . 7  |-  ( w  =  W  ->  (
(  _I  |`  s
) LIndF  w  <->  (  _I  |`  s
) LIndF  W ) )
53, 4rabeqbidv 2783 . . . . . 6  |-  ( w  =  W  ->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF  w }  =  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } )
6 df-linds 27277 . . . . . 6  |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s
) LIndF  w } )
7 fvex 5539 . . . . . . . 8  |-  ( Base `  W )  e.  _V
87pwex 4193 . . . . . . 7  |-  ~P ( Base `  W )  e. 
_V
98rabex 4165 . . . . . 6  |-  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W }  e.  _V
105, 6, 9fvmpt 5602 . . . . 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 2350 . . 3  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  X  e.  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } ) )
13 reseq2 4950 . . . . 5  |-  ( s  =  X  ->  (  _I  |`  s )  =  (  _I  |`  X ) )
1413breq1d 4033 . . . 4  |-  ( s  =  X  ->  (
(  _I  |`  s
) LIndF  W  <->  (  _I  |`  X ) LIndF 
W ) )
1514elrab 2923 . . 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 4175 . . . 4  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  ( Base `  W ) )
18 islinds.b . . . . 5  |-  B  =  ( Base `  W
)
1918sseq2i 3203 . . . 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   LIndF clindf 27274  LIndSclinds 27275
This theorem is referenced by:  linds1  27280  linds2  27281  islinds2  27283  lindsss  27294  lindsmm  27298  lsslinds  27301
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-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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-linds 27277
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