Theorem rellindf 21788

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5687. (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 21786	. 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 5776	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2114  ∀wral 3051  [wsbc 3728   ∖ cdif 3886  {csn 4567  dom cdm 5631   “ cima 5634  Rel wrel 5636  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402  LSpanclspn 20966   LIndF clindf 21784

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-opab 5148  df-xp 5637  df-rel 5638  df-lindf 21786

This theorem is referenced by:  lindff  21795  lindfind  21796  f1lindf  21802  lindfmm  21807  lsslindf  21810