Theorem rellindf 21775

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5688. (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 21773	. 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 5777	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  [wsbc 3742   ∖ cdif 3900  {csn 4582  dom cdm 5632   “ cima 5635  Rel wrel 5637  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Scalarcsca 17192   ·_𝑠 cvsca 17193  0_gc0g 17371  LSpanclspn 20934   LIndF clindf 21771

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-opab 5163  df-xp 5638  df-rel 5639  df-lindf 21773

This theorem is referenced by:  lindff  21782  lindfind  21783  f1lindf  21789  lindfmm  21794  lsslindf  21797