Theorem rellindf 21743

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5672. (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 21741	. 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 5760	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2111  ∀wral 3047  [wsbc 3741   ∖ cdif 3899  {csn 4576  dom cdm 5616   “ cima 5619  Rel wrel 5621  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  Scalarcsca 17161   ·_𝑠 cvsca 17162  0_gc0g 17340  LSpanclspn 20902   LIndF clindf 21739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3919  df-opab 5154  df-xp 5622  df-rel 5623  df-lindf 21741

This theorem is referenced by:  lindff  21750  lindfind  21751  f1lindf  21757  lindfmm  21762  lsslindf  21765