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Theorem lbsel 20922
Description: An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lbsss.v 𝑉 = (Baseβ€˜π‘Š)
lbsss.j 𝐽 = (LBasisβ€˜π‘Š)
Assertion
Ref Expression
lbsel ((𝐡 ∈ 𝐽 ∧ 𝐸 ∈ 𝐡) β†’ 𝐸 ∈ 𝑉)

Proof of Theorem lbsel
StepHypRef Expression
1 lbsss.v . . 3 𝑉 = (Baseβ€˜π‘Š)
2 lbsss.j . . 3 𝐽 = (LBasisβ€˜π‘Š)
31, 2lbsss 20921 . 2 (𝐡 ∈ 𝐽 β†’ 𝐡 βŠ† 𝑉)
43sselda 3975 1 ((𝐡 ∈ 𝐽 ∧ 𝐸 ∈ 𝐡) β†’ 𝐸 ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  β€˜cfv 6534  Basecbs 17149  LBasisclbs 20918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-lbs 20919
This theorem is referenced by:  lbsind2  20925
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