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Theorem rellindf 20502
 Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5573. (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 20500 . 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}))))}
21relopabi 5659 1 Rel LIndF
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  [wsbc 3720   ∖ cdif 3878  {csn 4525  dom cdm 5520   “ cima 5523  Rel wrel 5525  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  Scalarcsca 16563   ·𝑠 cvsca 16564  0gc0g 16708  LSpanclspn 19740   LIndF clindf 20498 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 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5094  df-xp 5526  df-rel 5527  df-lindf 20500 This theorem is referenced by:  lindff  20509  lindfind  20510  f1lindf  20516  lindfmm  20521  lsslindf  20524
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