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Theorem rellindf 20555
Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5408. (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 20553 . 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 5493 1 Rel LIndF
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 386  wcel 2107  wral 3090  [wsbc 3652  cdif 3789  {csn 4398  dom cdm 5357  cima 5360  Rel wrel 5362  wf 6133  cfv 6137  (class class class)co 6924  Basecbs 16259  Scalarcsca 16345   ·𝑠 cvsca 16346  0gc0g 16490  LSpanclspn 19370   LIndF clindf 20551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4951  df-xp 5363  df-rel 5364  df-lindf 20553
This theorem is referenced by:  lindff  20562  lindfind  20563  f1lindf  20569  lindfmm  20574  lsslindf  20577
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