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Theorem linds1 21830
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 ∈ (LIndS‘𝑊) → 𝑋𝐵)

Proof of Theorem linds1
StepHypRef Expression
1 elfvdm 6943 . . . 4 (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
2 islinds.b . . . . 5 𝐵 = (Base‘𝑊)
32islinds 21829 . . . 4 (𝑊 ∈ dom LIndS → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))
41, 3syl 17 . . 3 (𝑋 ∈ (LIndS‘𝑊) → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))
54ibi 267 . 2 (𝑋 ∈ (LIndS‘𝑊) → (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))
65simpld 494 1 (𝑋 ∈ (LIndS‘𝑊) → 𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951   class class class wbr 5143   I cid 5577  dom cdm 5685  cres 5687  cfv 6561  Basecbs 17247   LIndF clindf 21824  LIndSclinds 21825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-linds 21827
This theorem is referenced by:  lindsss  21844  lindsmm2  21849  islinds3  21854  islinds4  21855  0nellinds  33398  linds2eq  33409  lindsunlem  33675  lindsun  33676  dimkerim  33678  lindsadd  37620  lindsdom  37621  lindsenlbs  37622
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