Theorem rellindf 21733

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5679. (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 21731	. 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 5767	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  [wsbc 3744   ∖ cdif 3902  {csn 4579  dom cdm 5623   “ cima 5626  Rel wrel 5628  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  Scalarcsca 17182   ·_𝑠 cvsca 17183  0_gc0g 17361  LSpanclspn 20892   LIndF clindf 21729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-ss 3922  df-opab 5158  df-xp 5629  df-rel 5630  df-lindf 21731

This theorem is referenced by:  lindff  21740  lindfind  21741  f1lindf  21747  lindfmm  21752  lsslindf  21755