Theorem reldir 17233

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- U. U. R = U. U. R
2	1	isdir 17232	. . 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 ) ) ) ) )
3	2	ibi 256	. 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 ) ) ) )
4	3	simplld 791	1 |- ( R e. DirRel -> Rel R )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   U.cuni 4436   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   Rel wrel 5119   DirRelcdir 17228

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-dir 17230

This theorem is referenced by:  dirtr  17236  dirge  17237