Theorem rellindf 20364

Description: The independent-family predicate is a proper relation and can be used with brrelexi 5297. (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 20362	. 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 5383	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 382   ∈ wcel 2145  ∀wral 3061  [wsbc 3587   ∖ cdif 3720  {csn 4317  dom cdm 5250   “ cima 5253  Rel wrel 5255  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796  Basecbs 16064  Scalarcsca 16152   ·_𝑠 cvsca 16153  0_gc0g 16308  LSpanclspn 19184   LIndF clindf 20360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4848  df-xp 5256  df-rel 5257  df-lindf 20362

This theorem is referenced by:  lindff  20371  lindfind  20372  f1lindf  20378  lindfmm  20383  lsslindf  20386