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Theorem reldir 14641
Description: A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Assertion
Ref Expression
reldir  |-  ( R  e.  DirRel  ->  Rel  R )

Proof of Theorem reldir
StepHypRef Expression
1 eqid 2412 . . . . 5  |-  U. U. R  =  U. U. R
21isdir 14640 . . . 4  |-  ( R  e.  DirRel  ->  ( R  e. 
DirRel 
<->  ( ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R
)  /\  ( ( R  o.  R )  C_  R  /\  ( U. U. R  X.  U. U. R )  C_  ( `' R  o.  R
) ) ) ) )
32ibi 233 . . 3  |-  ( R  e.  DirRel  ->  ( ( Rel 
R  /\  (  _I  |` 
U. U. R )  C_  R )  /\  (
( R  o.  R
)  C_  R  /\  ( U. U. R  X.  U.
U. R )  C_  ( `' R  o.  R
) ) ) )
43simpld 446 . 2  |-  ( R  e.  DirRel  ->  ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R
) )
54simpld 446 1  |-  ( R  e.  DirRel  ->  Rel  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    C_ wss 3288   U.cuni 3983    _I cid 4461    X. cxp 4843   `'ccnv 4844    |` cres 4847    o. ccom 4849   Rel wrel 4850   DirRelcdir 14636
This theorem is referenced by:  dirtr  14644  dirge  14645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-v 2926  df-in 3295  df-ss 3302  df-uni 3984  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-res 4857  df-dir 14638
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