Theorem rellindf 21790

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5681. (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 21788	. 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 5770	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 396   ∈ wcel 2119  ∀wral 3054  [wsbc 3730   ∖ cdif 3887  {csn 4562  dom cdm 5625   “ cima 5628  Rel wrel 5630  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Scalarcsca 17221   ·_𝑠 cvsca 17222  0_gc0g 17400  LSpanclspn 20968   LIndF clindf 21786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-ss 3907  df-opab 5142  df-xp 5631  df-rel 5632  df-lindf 21788

This theorem is referenced by:  lindff  21797  lindfind  21798  f1lindf  21804  lindfmm  21809  lsslindf  21812