Theorem rellindf 20770

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5605. (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 20768	. 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 5690	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 399   ∈ wcel 2110  ∀wral 3061  [wsbc 3694   ∖ cdif 3863  {csn 4541  dom cdm 5551   “ cima 5554  Rel wrel 5556  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  Scalarcsca 16805   ·_𝑠 cvsca 16806  0_gc0g 16944  LSpanclspn 20008   LIndF clindf 20766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-in 3873  df-ss 3883  df-opab 5116  df-xp 5557  df-rel 5558  df-lindf 20768

This theorem is referenced by:  lindff  20777  lindfind  20778  f1lindf  20784  lindfmm  20789  lsslindf  20792