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Theorem linds2 19016
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 5874 . . . 4  |-  ( X  e.  (LIndS `  W
)  ->  W  e.  dom LIndS )
2 eqid 2454 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
32islinds 19014 . . . 4  |-  ( W  e.  dom LIndS  ->  ( X  e.  (LIndS `  W
)  <->  ( X  C_  ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W ) ) )
41, 3syl 16 . . 3  |-  ( X  e.  (LIndS `  W
)  ->  ( X  e.  (LIndS `  W )  <->  ( X  C_  ( Base `  W )  /\  (  _I  |`  X ) LIndF  W
) ) )
54ibi 241 . 2  |-  ( X  e.  (LIndS `  W
)  ->  ( X  C_  ( Base `  W
)  /\  (  _I  |`  X ) LIndF  W ) )
65simprd 461 1  |-  ( X  e.  (LIndS `  W
)  ->  (  _I  |`  X ) LIndF  W )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823    C_ wss 3461   class class class wbr 4439    _I cid 4779   dom cdm 4988    |` cres 4990   ` cfv 5570   Basecbs 14719   LIndF clindf 19009  LIndSclinds 19010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578  df-linds 19012
This theorem is referenced by:  lindsind2  19024  lindsss  19029  f1linds  19030
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