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Theorem rellindf 21230
Description: The independent-family predicate is a proper relation and can be used with brrelex1i 5689. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
rellindf Rel LIndF

Proof of Theorem rellindf
Dummy variables 𝑓 π‘˜ 𝑠 𝑀 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lindf 21228 . 2 LIndF = {βŸ¨π‘“, π‘€βŸ© ∣ (𝑓:dom π‘“βŸΆ(Baseβ€˜π‘€) ∧ [(Scalarβ€˜π‘€) / 𝑠]βˆ€π‘₯ ∈ dom π‘“βˆ€π‘˜ ∈ ((Baseβ€˜π‘ ) βˆ– {(0gβ€˜π‘ )}) Β¬ (π‘˜( ·𝑠 β€˜π‘€)(π‘“β€˜π‘₯)) ∈ ((LSpanβ€˜π‘€)β€˜(𝑓 β€œ (dom 𝑓 βˆ– {π‘₯}))))}
21relopabiv 5777 1 Rel LIndF
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3061  [wsbc 3740   βˆ– cdif 3908  {csn 4587  dom cdm 5634   β€œ cima 5637  Rel wrel 5639  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  LSpanclspn 20447   LIndF clindf 21226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-opab 5169  df-xp 5640  df-rel 5641  df-lindf 21228
This theorem is referenced by:  lindff  21237  lindfind  21238  f1lindf  21244  lindfmm  21249  lsslindf  21252
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