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Theorem linds1 21232
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 6880 . . . 4 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ π‘Š ∈ dom LIndS)
2 islinds.b . . . . 5 𝐡 = (Baseβ€˜π‘Š)
32islinds 21231 . . . 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 267 . 2 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ (𝑋 βŠ† 𝐡 ∧ ( I β†Ύ 𝑋) LIndF π‘Š))
65simpld 496 1 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ 𝑋 βŠ† 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911   class class class wbr 5106   I cid 5531  dom cdm 5634   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17088   LIndF clindf 21226  LIndSclinds 21227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-linds 21229
This theorem is referenced by:  lindsss  21246  lindsmm2  21251  islinds3  21256  islinds4  21257  0nellinds  32206  linds2eq  32216  lindsunlem  32376  lindsun  32377  dimkerim  32379  lindsadd  36117  lindsdom  36118  lindsenlbs  36119
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