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Theorem rellindf 21347
Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5730. (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 21345 . 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 5818 1 Rel LIndF
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wcel 2107  wral 3062  [wsbc 3776  cdif 3944  {csn 4627  dom cdm 5675  cima 5678  Rel wrel 5680  wf 6536  cfv 6540  (class class class)co 7404  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LSpanclspn 20570   LIndF clindf 21343
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-lindf 21345
This theorem is referenced by:  lindff  21354  lindfind  21355  f1lindf  21361  lindfmm  21366  lsslindf  21369
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