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Theorem linds2 20198
Description: An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
linds2 (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)

Proof of Theorem linds2
StepHypRef Expression
1 elfvdm 6258 . . . 4 (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
2 eqid 2651 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
32islinds 20196 . . . 4 (𝑊 ∈ dom LIndS → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)))
41, 3syl 17 . . 3 (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊)))
54ibi 256 . 2 (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ⊆ (Base‘𝑊) ∧ ( I ↾ 𝑋) LIndF 𝑊))
65simprd 478 1 (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  wss 3607   class class class wbr 4685   I cid 5052  dom cdm 5143  cres 5145  cfv 5926  Basecbs 15904   LIndF clindf 20191  LIndSclinds 20192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-linds 20194
This theorem is referenced by:  lindsind2  20206  lindsss  20211  f1linds  20212
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