Theorem rellindf 22147

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5712. (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 22145	. 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	relopabi 5806	1 ⊢ Rel LIndF

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∈ wcel 2147  ∀wral 3080  [wsbc 3742   ∖ cdif 4048  {csn 4588  dom cdm 5656   “ cima 5659  Rel wrel 5661  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Scalarcsca 17463   ·_𝑠 cvsca 17464  0_gc0g 17642  LSpanclspn 21279   LIndF clindf 22143

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5172  df-xp 5662  df-rel 5663  df-lindf 22145

This theorem is used by:  lindff  22154  lindfind  22155  f1lindf  22161  lindfmm  22166  lsslindf  22169