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Theorem linds1 27257
Description: An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
islinds.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
linds1  |-  ( X  e.  (LIndS `  W
)  ->  X  C_  B
)

Proof of Theorem linds1
StepHypRef Expression
1 elfvdm 5757 . . . 4  |-  ( X  e.  (LIndS `  W
)  ->  W  e.  dom LIndS )
2 islinds.b . . . . 5  |-  B  =  ( Base `  W
)
32islinds 27256 . . . 4  |-  ( W  e.  dom LIndS  ->  ( X  e.  (LIndS `  W
)  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF 
W ) ) )
41, 3syl 16 . . 3  |-  ( X  e.  (LIndS `  W
)  ->  ( X  e.  (LIndS `  W )  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF 
W ) ) )
54ibi 233 . 2  |-  ( X  e.  (LIndS `  W
)  ->  ( X  C_  B  /\  (  _I  |`  X ) LIndF  W ) )
65simpld 446 1  |-  ( X  e.  (LIndS `  W
)  ->  X  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   class class class wbr 4212    _I cid 4493   dom cdm 4878    |` cres 4880   ` cfv 5454   Basecbs 13469   LIndF clindf 27251  LIndSclinds 27252
This theorem is referenced by:  lindsss  27271  lindsmm2  27276  islinds3  27281  islinds4  27282
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fv 5462  df-linds 27254
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