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Theorem linds2 21702
Description: An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
linds2 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ ( I β†Ύ 𝑋) LIndF π‘Š)

Proof of Theorem linds2
StepHypRef Expression
1 elfvdm 6921 . . . 4 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ π‘Š ∈ dom LIndS)
2 eqid 2726 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
32islinds 21700 . . . 4 (π‘Š ∈ dom LIndS β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 βŠ† (Baseβ€˜π‘Š) ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))
41, 3syl 17 . . 3 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ (𝑋 ∈ (LIndSβ€˜π‘Š) ↔ (𝑋 βŠ† (Baseβ€˜π‘Š) ∧ ( I β†Ύ 𝑋) LIndF π‘Š)))
54ibi 267 . 2 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ (𝑋 βŠ† (Baseβ€˜π‘Š) ∧ ( I β†Ύ 𝑋) LIndF π‘Š))
65simprd 495 1 (𝑋 ∈ (LIndSβ€˜π‘Š) β†’ ( I β†Ύ 𝑋) LIndF π‘Š)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098   βŠ† wss 3943   class class class wbr 5141   I cid 5566  dom cdm 5669   β†Ύ cres 5671  β€˜cfv 6536  Basecbs 17151   LIndF clindf 21695  LIndSclinds 21696
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-linds 21698
This theorem is referenced by:  lindsind2  21710  lindsss  21715  f1linds  21716
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