Theorem rellindf 20147

Description: The independent-family predicate is a proper relation and can be used with brrelexi 5158. (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 20145	. 2 |- LIndF = { <. f , w >. | ( f : dom f --> ( Base ` w ) /\ [. (Scalar ` w ) / s ]. A. x e. dom f A. k e. ( ( Base ` s ) \ { ( 0g ` s ) } ) -. ( k ( .s ` w ) ( f ` x ) ) e. ( ( LSpan ` w ) ` ( f " ( dom f \ { x } ) ) ) ) }
2	1	relopabi 5245	1 |- Rel LIndF

Syntax hints:   -. wn 3   /\ wa 384   e. wcel 1990   A.wral 2912   [.wsbc 3435   \ cdif 3571   {csn 4177   dom cdm 5114   "cima 5117   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LSpanclspn 18971   LIndF clindf 20143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-lindf 20145

This theorem is referenced by:  lindff  20154  lindfind  20155  f1lindf  20161  lindfmm  20166  lsslindf  20169