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Theorem linds1 20497
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 6684 . . . 4 (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ dom LIndS)
2 islinds.b . . . . 5 𝐵 = (Base‘𝑊)
32islinds 20496 . . . 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 270 . 2 (𝑋 ∈ (LIndS‘𝑊) → (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))
65simpld 498 1 (𝑋 ∈ (LIndS‘𝑊) → 𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wss 3908   class class class wbr 5042   I cid 5436  dom cdm 5532  cres 5534  cfv 6334  Basecbs 16474   LIndF clindf 20491  LIndSclinds 20492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-res 5544  df-iota 6293  df-fun 6336  df-fv 6342  df-linds 20494
This theorem is referenced by:  lindsss  20511  lindsmm2  20516  islinds3  20521  islinds4  20522  0nellinds  30967  linds2eq  30976  lindsunlem  31077  lindsun  31078  dimkerim  31080  lindsadd  35008  lindsdom  35009  lindsenlbs  35010
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