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Theorem dirref 16189
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 2402 . . . 4  |-  A  =  A
2 resieq 5104 . . . . 5  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A (  _I  |`  X ) A  <->  A  =  A ) )
32anidms 643 . . . 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 16188 . . . . . . 7  |-  ( R  e.  DirRel  ->  dom  R  =  U. U. R )
75, 6syl5eq 2455 . . . . . 6  |-  ( R  e.  DirRel  ->  X  =  U. U. R )
87reseq2d 5094 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  X )  =  (  _I  |`  U. U. R ) )
9 eqid 2402 . . . . . . . . 9  |-  U. U. R  =  U. U. R
109isdir 16186 . . . . . . . 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 457 . . . . . 6  |-  ( R  e.  DirRel  ->  ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R
) )
1312simprd 461 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  U. U. R )  C_  R
)
148, 13eqsstrd 3476 . . . 4  |-  ( R  e.  DirRel  ->  (  _I  |`  X ) 
C_  R )
1514ssbrd 4436 . . 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 427 1  |-  ( ( R  e.  DirRel  /\  A  e.  X )  ->  A R A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   U.cuni 4191   class class class wbr 4395    _I cid 4733    X. cxp 4821   `'ccnv 4822   dom cdm 4823    |` cres 4825    o. ccom 4827   Rel wrel 4828   DirRelcdir 16182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-dir 16184
This theorem is referenced by:  tailini  30604
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