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Theorem rellindf 21846
Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5745. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
rellindf Rel LIndF

Proof of Theorem rellindf
Dummy variables 𝑓 𝑘 𝑠 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lindf 21844 . 2 LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
21relopabiv 5833 1 Rel LIndF
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wcel 2106  wral 3059  [wsbc 3791  cdif 3960  {csn 4631  dom cdm 5689  cima 5692  Rel wrel 5694  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486  LSpanclspn 20987   LIndF clindf 21842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696  df-lindf 21844
This theorem is referenced by:  lindff  21853  lindfind  21854  f1lindf  21860  lindfmm  21865  lsslindf  21868
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