Theorem rellindf 22119

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5706. (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 22117	. 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 5797	1 ⊢ Rel LIndF

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∈ wcel 2144  ∀wral 3076  [wsbc 3738   ∖ cdif 4045  {csn 4584  dom cdm 5650   “ cima 5653  Rel wrel 5655  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  Scalarcsca 17436   ·_𝑠 cvsca 17437  0_gc0g 17615  LSpanclspn 21251   LIndF clindf 22115

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-opab 5167  df-xp 5656  df-rel 5657  df-lindf 22117

This theorem is used by:  lindff  22126  lindfind  22127  f1lindf  22133  lindfmm  22138  lsslindf  22141