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Theorem reldir 14683
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 2438 . . . . 5  |-  U. U. R  =  U. U. R
21isdir 14682 . . . 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 234 . . 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 447 . 2  |-  ( R  e.  DirRel  ->  ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R
) )
54simpld 447 1  |-  ( R  e.  DirRel  ->  Rel  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    C_ wss 3322   U.cuni 4017    _I cid 4496    X. cxp 4879   `'ccnv 4880    |` cres 4883    o. ccom 4885   Rel wrel 4886   DirRelcdir 14678
This theorem is referenced by:  dirtr  14686  dirge  14687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-uni 4018  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-res 4893  df-dir 14680
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