Theorem rellindf 21363

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5733. (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 21361	. 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 5821	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 397   ∈ wcel 2107  ∀wral 3062  [wsbc 3778   ∖ cdif 3946  {csn 4629  dom cdm 5677   “ cima 5680  Rel wrel 5682  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200   ·_𝑠 cvsca 17201  0_gc0g 17385  LSpanclspn 20582   LIndF clindf 21359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-lindf 21361

This theorem is referenced by:  lindff  21370  lindfind  21371  f1lindf  21377  lindfmm  21382  lsslindf  21385