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Theorem dirref 15711
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 2460 . . . 4  |-  A  =  A
2 resieq 5275 . . . . 5  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A (  _I  |`  X ) A  <->  A  =  A ) )
32anidms 645 . . . 4  |-  ( A  e.  X  ->  ( A (  _I  |`  X ) A  <->  A  =  A
) )
41, 3mpbiri 233 . . 3  |-  ( A  e.  X  ->  A
(  _I  |`  X ) A )
5 dirref.1 . . . . . . 7  |-  X  =  dom  R
6 dirdm 15710 . . . . . . 7  |-  ( R  e.  DirRel  ->  dom  R  =  U. U. R )
75, 6syl5eq 2513 . . . . . 6  |-  ( R  e.  DirRel  ->  X  =  U. U. R )
87reseq2d 5264 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  X )  =  (  _I  |`  U. U. R ) )
9 eqid 2460 . . . . . . . . 9  |-  U. U. R  =  U. U. R
109isdir 15708 . . . . . . . 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 241 . . . . . . 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 459 . . . . . 6  |-  ( R  e.  DirRel  ->  ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R
) )
1312simprd 463 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  U. U. R )  C_  R
)
148, 13eqsstrd 3531 . . . 4  |-  ( R  e.  DirRel  ->  (  _I  |`  X ) 
C_  R )
1514ssbrd 4481 . . 3  |-  ( R  e.  DirRel  ->  ( A (  _I  |`  X ) A  ->  A R A ) )
164, 15syl5 32 . 2  |-  ( R  e.  DirRel  ->  ( A  e.  X  ->  A R A ) )
1716imp 429 1  |-  ( ( R  e.  DirRel  /\  A  e.  X )  ->  A R A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3469   U.cuni 4238   class class class wbr 4440    _I cid 4783    X. cxp 4990   `'ccnv 4991   dom cdm 4992    |` cres 4994    o. ccom 4996   Rel wrel 4997   DirRelcdir 15704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-dir 15706
This theorem is referenced by:  tailini  29648
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