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Theorem dirref 14373
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 2296 . . . 4  |-  A  =  A
2 resieq 4981 . . . . 5  |-  ( ( A  e.  X  /\  A  e.  X )  ->  ( A (  _I  |`  X ) A  <->  A  =  A ) )
32anidms 626 . . . 4  |-  ( A  e.  X  ->  ( A (  _I  |`  X ) A  <->  A  =  A
) )
41, 3mpbiri 224 . . 3  |-  ( A  e.  X  ->  A
(  _I  |`  X ) A )
5 dirref.1 . . . . . . 7  |-  X  =  dom  R
6 dirdm 14372 . . . . . . 7  |-  ( R  e.  DirRel  ->  dom  R  =  U. U. R )
75, 6syl5eq 2340 . . . . . 6  |-  ( R  e.  DirRel  ->  X  =  U. U. R )
87reseq2d 4971 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  X )  =  (  _I  |`  U. U. R ) )
9 eqid 2296 . . . . . . . . 9  |-  U. U. R  =  U. U. R
109isdir 14370 . . . . . . . 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 232 . . . . . . 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 445 . . . . . 6  |-  ( R  e.  DirRel  ->  ( Rel  R  /\  (  _I  |`  U. U. R )  C_  R
) )
1312simprd 449 . . . . 5  |-  ( R  e.  DirRel  ->  (  _I  |`  U. U. R )  C_  R
)
148, 13eqsstrd 3225 . . . 4  |-  ( R  e.  DirRel  ->  (  _I  |`  X ) 
C_  R )
1514ssbrd 4080 . . 3  |-  ( R  e.  DirRel  ->  ( A (  _I  |`  X ) A  ->  A R A ) )
164, 15syl5 28 . 2  |-  ( R  e.  DirRel  ->  ( A  e.  X  ->  A R A ) )
1716imp 418 1  |-  ( ( R  e.  DirRel  /\  A  e.  X )  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   class class class wbr 4039    _I cid 4320    X. cxp 4703   `'ccnv 4704   dom cdm 4705    |` cres 4707    o. ccom 4709   Rel wrel 4710   DirRelcdir 14366
This theorem is referenced by:  tailini  26428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-dir 14368
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