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Theorem dirref 14668
Description: A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Hypothesis
Ref Expression
dirref.1  |-  X  =  dom  R
Assertion
Ref Expression
dirref  |-  ( ( R  e.  DirRel  /\  A  e.  X )  ->  A R A )

Proof of Theorem dirref
StepHypRef Expression
1 eqid 2435 . . . 4  |-  A  =  A
2 resieq 5147 . . . . 5  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A (  _I  |`  X ) A  <->  A  =  A ) )
32anidms 627 . . . 4  |-  ( A  e.  X  ->  ( A (  _I  |`  X ) A  <->  A  =  A
) )
41, 3mpbiri 225 . . 3  |-  ( A  e.  X  ->  A
(  _I  |`  X ) A )
5 dirref.1 . . . . . . 7  |-  X  =  dom  R
6 dirdm 14667 . . . . . . 7  |-  ( R  e.  DirRel  ->  dom  R  =  U. U. R )
75, 6syl5eq 2479 . . . . . 6  |-  ( R  e.  DirRel  ->  X  =  U. U. R )
87reseq2d 5137 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  X )  =  (  _I  |`  U. U. R ) )
9 eqid 2435 . . . . . . . . 9  |-  U. U. R  =  U. U. R
109isdir 14665 . . . . . . . 8  |-  ( 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
) ) ) ) )
1110ibi 233 . . . . . . 7  |-  ( 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
) ) ) )
1211simpld 446 . . . . . 6  |-  ( R  e.  DirRel  ->  ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R
) )
1312simprd 450 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  U. U. R )  C_  R
)
148, 13eqsstrd 3374 . . . 4  |-  ( R  e.  DirRel  ->  (  _I  |`  X ) 
C_  R )
1514ssbrd 4245 . . 3  |-  ( R  e.  DirRel  ->  ( A (  _I  |`  X ) A  ->  A R A ) )
164, 15syl5 30 . 2  |-  ( R  e.  DirRel  ->  ( A  e.  X  ->  A R A ) )
1716imp 419 1  |-  ( ( R  e.  DirRel  /\  A  e.  X )  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   U.cuni 4007   class class class wbr 4204    _I cid 4485    X. cxp 4867   `'ccnv 4868   dom cdm 4869    |` cres 4871    o. ccom 4873   Rel wrel 4874   DirRelcdir 14661
This theorem is referenced by:  tailini  26342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-dir 14663
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