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Theorem islinds 19014
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 3115 . . . . 5  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5848 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
32pweqd 4004 . . . . . . 7  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
4 breq2 4443 . . . . . . 7  |-  ( w  =  W  ->  (
(  _I  |`  s
) LIndF  w  <->  (  _I  |`  s
) LIndF  W ) )
53, 4rabeqbidv 3101 . . . . . 6  |-  ( w  =  W  ->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF  w }  =  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } )
6 df-linds 19012 . . . . . 6  |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s
) LIndF  w } )
7 fvex 5858 . . . . . . . 8  |-  ( Base `  W )  e.  _V
87pwex 4620 . . . . . . 7  |-  ~P ( Base `  W )  e. 
_V
98rabex 4588 . . . . . 6  |-  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W }  e.  _V
105, 6, 9fvmpt 5931 . . . . 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 2524 . . 3  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  X  e.  { s  e.  ~P ( Base `  W )  |  (  _I  |`  s ) LIndF  W } ) )
13 reseq2 5257 . . . . 5  |-  ( s  =  X  ->  (  _I  |`  s )  =  (  _I  |`  X ) )
1413breq1d 4449 . . . 4  |-  ( s  =  X  ->  (
(  _I  |`  s
) LIndF  W  <->  (  _I  |`  X ) LIndF 
W ) )
1514elrab 3254 . . 3  |-  ( X  e.  { s  e. 
~P ( Base `  W
)  |  (  _I  |`  s ) LIndF  W }  <->  ( X  e.  ~P ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W ) )
1612, 15syl6bb 261 . 2  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  ( X  e. 
~P ( Base `  W
)  /\  (  _I  |`  X ) LIndF  W ) ) )
177elpw2 4601 . . . 4  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  ( Base `  W ) )
18 islinds.b . . . . 5  |-  B  =  ( Base `  W
)
1918sseq2i 3514 . . . 4  |-  ( X 
C_  B  <->  X  C_  ( Base `  W ) )
2017, 19bitr4i 252 . . 3  |-  ( X  e.  ~P ( Base `  W )  <->  X  C_  B
)
2120anbi1i 693 . 2  |-  ( ( X  e.  ~P ( Base `  W )  /\  (  _I  |`  X ) LIndF 
W )  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF  W ) )
2216, 21syl6bb 261 1  |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W
)  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF 
W ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   class class class wbr 4439    _I cid 4779    |` 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-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:  linds1  19015  linds2  19016  islinds2  19018  lindsss  19029  lindsmm  19033  lsslinds  19036
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