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Theorem rellindf 20369
 Description: The independent-family predicate is a proper relation and can be used with brrelexi 5315. (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 20367 . 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 5401 1 Rel LIndF
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   ∈ wcel 2139  ∀wral 3050  [wsbc 3576   ∖ cdif 3712  {csn 4321  dom cdm 5266   “ cima 5269  Rel wrel 5271  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814  Basecbs 16079  Scalarcsca 16166   ·𝑠 cvsca 16167  0gc0g 16322  LSpanclspn 19193   LIndF clindf 20365 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273  df-lindf 20367 This theorem is referenced by:  lindff  20376  lindfind  20377  f1lindf  20383  lindfmm  20388  lsslindf  20391
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