Theorem rellindf 20555

Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5408. (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 20553	. 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 5493	1 ⊢ Rel LIndF

Syntax hints:  ¬ wn 3   ∧ wa 386   ∈ wcel 2107  ∀wral 3090  [wsbc 3652   ∖ cdif 3789  {csn 4398  dom cdm 5357   “ cima 5360  Rel wrel 5362  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  Basecbs 16259  Scalarcsca 16345   ·_𝑠 cvsca 16346  0_gc0g 16490  LSpanclspn 19370   LIndF clindf 20551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-opab 4951  df-xp 5363  df-rel 5364  df-lindf 20553

This theorem is referenced by:  lindff  20562  lindfind  20563  f1lindf  20569  lindfmm  20574  lsslindf  20577