Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  linds2 Unicode version

Theorem linds2 27384
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 5570 . . . 4  |-  ( X  e.  (LIndS `  W
)  ->  W  e.  dom LIndS )
2 eqid 2296 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
32islinds 27382 . . . 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 1696    C_ wss 3165   class class class wbr 4039    _I cid 4320   dom cdm 4705    |` cres 4707   ` cfv 5271   Basecbs 13164   LIndF clindf 27377  LIndSclinds 27378
This theorem is referenced by:  lindsind2  27392  lindsss  27397  f1linds  27398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-linds 27380
  Copyright terms: Public domain W3C validator