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Theorem isdir 14677
Description: A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Hypothesis
Ref Expression
isdir.1  |-  A  = 
U. U. R
Assertion
Ref Expression
isdir  |-  ( R  e.  V  ->  ( R  e.  DirRel  <->  ( ( Rel  R  /\  (  _I  |`  A )  C_  R
)  /\  ( ( R  o.  R )  C_  R  /\  ( A  X.  A )  C_  ( `' R  o.  R
) ) ) ) )

Proof of Theorem isdir
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 releq 4959 . . . 4  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 unieq 4024 . . . . . . . 8  |-  ( r  =  R  ->  U. r  =  U. R )
32unieqd 4026 . . . . . . 7  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
4 isdir.1 . . . . . . 7  |-  A  = 
U. U. R
53, 4syl6eqr 2486 . . . . . 6  |-  ( r  =  R  ->  U. U. r  =  A )
65reseq2d 5146 . . . . 5  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  A ) )
7 id 20 . . . . 5  |-  ( r  =  R  ->  r  =  R )
86, 7sseq12d 3377 . . . 4  |-  ( r  =  R  ->  (
(  _I  |`  U. U. r )  C_  r  <->  (  _I  |`  A )  C_  R ) )
91, 8anbi12d 692 . . 3  |-  ( r  =  R  ->  (
( Rel  r  /\  (  _I  |`  U. U. r )  C_  r
)  <->  ( Rel  R  /\  (  _I  |`  A ) 
C_  R ) ) )
107, 7coeq12d 5037 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
1110, 7sseq12d 3377 . . . 4  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
125, 5xpeq12d 4903 . . . . 5  |-  ( r  =  R  ->  ( U. U. r  X.  U. U. r )  =  ( A  X.  A ) )
13 cnveq 5046 . . . . . 6  |-  ( r  =  R  ->  `' r  =  `' R
)
1413, 7coeq12d 5037 . . . . 5  |-  ( r  =  R  ->  ( `' r  o.  r
)  =  ( `' R  o.  R ) )
1512, 14sseq12d 3377 . . . 4  |-  ( r  =  R  ->  (
( U. U. r  X.  U. U. r ) 
C_  ( `' r  o.  r )  <->  ( A  X.  A )  C_  ( `' R  o.  R
) ) )
1611, 15anbi12d 692 . . 3  |-  ( r  =  R  ->  (
( ( r  o.  r )  C_  r  /\  ( U. U. r  X.  U. U. r ) 
C_  ( `' r  o.  r ) )  <-> 
( ( R  o.  R )  C_  R  /\  ( A  X.  A
)  C_  ( `' R  o.  R )
) ) )
179, 16anbi12d 692 . 2  |-  ( r  =  R  ->  (
( ( Rel  r  /\  (  _I  |`  U. U. r )  C_  r
)  /\  ( (
r  o.  r ) 
C_  r  /\  ( U. U. r  X.  U. U. r )  C_  ( `' r  o.  r
) ) )  <->  ( ( Rel  R  /\  (  _I  |`  A )  C_  R
)  /\  ( ( R  o.  R )  C_  R  /\  ( A  X.  A )  C_  ( `' R  o.  R
) ) ) ) )
18 df-dir 14675 . 2  |-  DirRel  =  {
r  |  ( ( Rel  r  /\  (  _I  |`  U. U. r
)  C_  r )  /\  ( ( r  o.  r )  C_  r  /\  ( U. U. r  X.  U. U. r ) 
C_  ( `' r  o.  r ) ) ) }
1917, 18elab2g 3084 1  |-  ( R  e.  V  ->  ( R  e.  DirRel  <->  ( ( Rel  R  /\  (  _I  |`  A )  C_  R
)  /\  ( ( R  o.  R )  C_  R  /\  ( A  X.  A )  C_  ( `' R  o.  R
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   U.cuni 4015    _I cid 4493    X. cxp 4876   `'ccnv 4877    |` cres 4880    o. ccom 4882   Rel wrel 4883   DirRelcdir 14673
This theorem is referenced by:  reldir  14678  dirdm  14679  dirref  14680  dirtr  14681  dirge  14682  tsrdir  14683  filnetlem3  26409
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-res 4890  df-dir 14675
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