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Theorem linds2 27281
Description: An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
linds2  |-  ( X  e.  (LIndS `  W
)  ->  (  _I  |`  X ) LIndF  W )

Proof of Theorem linds2
StepHypRef Expression
1 elfvdm 5554 . . . 4  |-  ( X  e.  (LIndS `  W
)  ->  W  e.  dom LIndS )
2 eqid 2283 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
32islinds 27279 . . . 4  |-  ( W  e.  dom LIndS  ->  ( X  e.  (LIndS `  W
)  <->  ( X  C_  ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W ) ) )
41, 3syl 15 . . 3  |-  ( X  e.  (LIndS `  W
)  ->  ( X  e.  (LIndS `  W )  <->  ( X  C_  ( Base `  W )  /\  (  _I  |`  X ) LIndF  W
) ) )
54ibi 232 . 2  |-  ( X  e.  (LIndS `  W
)  ->  ( X  C_  ( Base `  W
)  /\  (  _I  |`  X ) LIndF  W ) )
65simprd 449 1  |-  ( X  e.  (LIndS `  W
)  ->  (  _I  |`  X ) LIndF  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    C_ wss 3152   class class class wbr 4023    _I cid 4304   dom cdm 4689    |` cres 4691   ` cfv 5255   Basecbs 13148   LIndF clindf 27274  LIndSclinds 27275
This theorem is referenced by:  lindsind2  27289  lindsss  27294  f1linds  27295
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
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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