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Theorem islinds 27247
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 2956 . . . . 5  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5720 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32pweqd 3796 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
4 breq2 4208 . . . . . . 7  |-  ( w  =  W  ->  (
(  _I  |`  s
) LIndF  w  <->  (  _I  |`  s
) LIndF  W ) )
53, 4rabeqbidv 2943 . . . . . 6  |-  ( w  =  W  ->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF  w }  =  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } )
6 df-linds 27245 . . . . . 6  |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s
) LIndF  w } )
7 fvex 5734 . . . . . . . 8  |-  ( Base `  W )  e.  _V
87pwex 4374 . . . . . . 7  |-  ~P ( Base `  W )  e. 
_V
98rabex 4346 . . . . . 6  |-  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W }  e.  _V
105, 6, 9fvmpt 5798 . . . . 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 2502 . . 3  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  X  e.  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } ) )
13 reseq2 5133 . . . . 5  |-  ( s  =  X  ->  (  _I  |`  s )  =  (  _I  |`  X ) )
1413breq1d 4214 . . . 4  |-  ( s  =  X  ->  (
(  _I  |`  s
) LIndF  W  <->  (  _I  |`  X ) LIndF 
W ) )
1514elrab 3084 . . 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 4356 . . . 4  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  ( Base `  W ) )
18 islinds.b . . . . 5  |-  B  =  ( Base `  W
)
1918sseq2i 3365 . . . 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   class class class wbr 4204    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   LIndF clindf 27242  LIndSclinds 27243
This theorem is referenced by:  linds1  27248  linds2  27249  islinds2  27251  lindsss  27262  lindsmm  27266  lsslinds  27269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-linds 27245
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