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Theorem islinds 26948
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 2907 . . . . 5  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5668 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32pweqd 3747 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
4 breq2 4157 . . . . . . 7  |-  ( w  =  W  ->  (
(  _I  |`  s
) LIndF  w  <->  (  _I  |`  s
) LIndF  W ) )
53, 4rabeqbidv 2894 . . . . . 6  |-  ( w  =  W  ->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF  w }  =  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } )
6 df-linds 26946 . . . . . 6  |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s
) LIndF  w } )
7 fvex 5682 . . . . . . . 8  |-  ( Base `  W )  e.  _V
87pwex 4323 . . . . . . 7  |-  ~P ( Base `  W )  e. 
_V
98rabex 4295 . . . . . 6  |-  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W }  e.  _V
105, 6, 9fvmpt 5745 . . . . 5  |-  ( W  e.  _V  ->  (LIndS `  W )  =  {
s  e.  ~P ( Base `  W )  |  (  _I  |`  s
) LIndF  W } )
111, 10syl 16 . . . 4  |-  ( W  e.  V  ->  (LIndS `  W )  =  {
s  e.  ~P ( Base `  W )  |  (  _I  |`  s
) LIndF  W } )
1211eleq2d 2454 . . 3  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  X  e.  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } ) )
13 reseq2 5081 . . . . 5  |-  ( s  =  X  ->  (  _I  |`  s )  =  (  _I  |`  X ) )
1413breq1d 4163 . . . 4  |-  ( s  =  X  ->  (
(  _I  |`  s
) LIndF  W  <->  (  _I  |`  X ) LIndF 
W ) )
1514elrab 3035 . . 3  |-  ( X  e.  { s  e. 
~P ( Base `  W
)  |  (  _I  |`  s ) LIndF  W }  <->  ( X  e.  ~P ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W ) )
1612, 15syl6bb 253 . 2  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  ( X  e. 
~P ( Base `  W
)  /\  (  _I  |`  X ) LIndF  W ) ) )
177elpw2 4305 . . . 4  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  ( Base `  W ) )
18 islinds.b . . . . 5  |-  B  =  ( Base `  W
)
1918sseq2i 3316 . . . 4  |-  ( X 
C_  B  <->  X  C_  ( Base `  W ) )
2017, 19bitr4i 244 . . 3  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  B
)
2120anbi1i 677 . 2  |-  ( ( X  e.  ~P ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W )  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF  W ) )
2216, 21syl6bb 253 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2653   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   class class class wbr 4153    _I cid 4434    |` cres 4820   ` cfv 5394   Basecbs 13396   LIndF clindf 26943  LIndSclinds 26944
This theorem is referenced by:  linds1  26949  linds2  26950  islinds2  26952  lindsss  26963  lindsmm  26967  lsslinds  26970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-iota 5358  df-fun 5396  df-fv 5402  df-linds 26946
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