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Theorem dirref 15504
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 2451 . . . 4  |-  A  =  A
2 resieq 5216 . . . . 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 15503 . . . . . . 7  |-  ( R  e.  DirRel  ->  dom  R  =  U. U. R )
75, 6syl5eq 2503 . . . . . 6  |-  ( R  e.  DirRel  ->  X  =  U. U. R )
87reseq2d 5205 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  X )  =  (  _I  |`  U. U. R ) )
9 eqid 2451 . . . . . . . . 9  |-  U. U. R  =  U. U. R
109isdir 15501 . . . . . . . 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 3485 . . . 4  |-  ( R  e.  DirRel  ->  (  _I  |`  X ) 
C_  R )
1514ssbrd 4428 . . 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 1370    e. wcel 1758    C_ wss 3423   U.cuni 4186   class class class wbr 4387    _I cid 4726    X. cxp 4933   `'ccnv 4934   dom cdm 4935    |` cres 4937    o. ccom 4939   Rel wrel 4940   DirRelcdir 15497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-dir 15499
This theorem is referenced by:  tailini  28732
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