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Theorem reldir 15990
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 2457 . . . 4  |-  U. U. R  =  U. U. R
21isdir 15989 . . 3  |-  ( 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 241 . 2  |-  ( 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
) ) ) )
43simplld 754 1  |-  ( R  e.  DirRel  ->  Rel  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819    C_ wss 3471   U.cuni 4251    _I cid 4799    X. cxp 5006   `'ccnv 5007    |` cres 5010    o. ccom 5012   Rel wrel 5013   DirRelcdir 15985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-res 5020  df-dir 15987
This theorem is referenced by:  dirtr  15993  dirge  15994
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