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Theorem linds1 21751
Description: An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypothesis
Ref Expression
islinds.b 𝐡 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
linds1 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ 𝑋 βŠ† 𝐡)

Proof of Theorem linds1
StepHypRef Expression
1 elfvdm 6939 . . . 4 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ π‘Š ∈ dom LIndS)
2 islinds.b . . . . 5 𝐡 = (Baseβ€˜π‘Š)
32islinds 21750 . . . 4 (π‘Š ∈ dom LIndS β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 βŠ† 𝐡 ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))
41, 3syl 17 . . 3 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 βŠ† 𝐡 ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))
54ibi 266 . 2 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ (𝑋 βŠ† 𝐡 ∧ ( I β†Ύ 𝑋) LIndF π‘Š))
65simpld 493 1 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ 𝑋 βŠ† 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949   class class class wbr 5152   I cid 5579  dom cdm 5682   β†Ύ cres 5684  β€˜cfv 6553  Basecbs 17187   LIndF clindf 21745  LIndSclinds 21746
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-linds 21748
This theorem is referenced by:  lindsss  21765  lindsmm2  21770  islinds3  21775  islinds4  21776  0nellinds  33106  linds2eq  33121  lindsunlem  33355  lindsun  33356  dimkerim  33358  lindsadd  37119  lindsdom  37120  lindsenlbs  37121
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