Theorem rellindf 21369

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5732. (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.

Step	Hyp	Ref	Expression
1		df-lindf 21367	. 2 ⊢ LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
2	1	relopabiv 5820	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 396   ∈ wcel 2106  ∀wral 3061  [wsbc 3777   ∖ cdif 3945  {csn 4628  dom cdm 5676   “ cima 5679  Rel wrel 5681  ⟶wf 6539  ‘cfv 6543  (class class class)co 7411  Basecbs 17146  Scalarcsca 17202   ·_𝑠 cvsca 17203  0_gc0g 17387  LSpanclspn 20587   LIndF clindf 21365

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683  df-lindf 21367

This theorem is referenced by:  lindff  21376  lindfind  21377  f1lindf  21383  lindfmm  21388  lsslindf  21391