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Theorem isdir 15524
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 5033 . . . 4  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 unieq 4210 . . . . . . . 8  |-  ( r  =  R  ->  U. r  =  U. R )
32unieqd 4212 . . . . . . 7  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
4 isdir.1 . . . . . . 7  |-  A  = 
U. U. R
53, 4syl6eqr 2513 . . . . . 6  |-  ( r  =  R  ->  U. U. r  =  A )
65reseq2d 5221 . . . . 5  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  A ) )
7 id 22 . . . . 5  |-  ( r  =  R  ->  r  =  R )
86, 7sseq12d 3496 . . . 4  |-  ( r  =  R  ->  (
(  _I  |`  U. U. r )  C_  r  <->  (  _I  |`  A )  C_  R ) )
91, 8anbi12d 710 . . 3  |-  ( r  =  R  ->  (
( Rel  r  /\  (  _I  |`  U. U. r )  C_  r
)  <->  ( Rel  R  /\  (  _I  |`  A ) 
C_  R ) ) )
107, 7coeq12d 5115 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
1110, 7sseq12d 3496 . . . 4  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
125, 5xpeq12d 4976 . . . . 5  |-  ( r  =  R  ->  ( U. U. r  X.  U. U. r )  =  ( A  X.  A ) )
13 cnveq 5124 . . . . . 6  |-  ( r  =  R  ->  `' r  =  `' R
)
1413, 7coeq12d 5115 . . . . 5  |-  ( r  =  R  ->  ( `' r  o.  r
)  =  ( `' R  o.  R ) )
1512, 14sseq12d 3496 . . . 4  |-  ( r  =  R  ->  (
( U. U. r  X.  U. U. r ) 
C_  ( `' r  o.  r )  <->  ( A  X.  A )  C_  ( `' R  o.  R
) ) )
1611, 15anbi12d 710 . . 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 710 . 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 15522 . 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 3215 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3439   U.cuni 4202    _I cid 4742    X. cxp 4949   `'ccnv 4950    |` cres 4953    o. ccom 4955   Rel wrel 4956   DirRelcdir 15520
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-v 3080  df-in 3446  df-ss 3453  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-res 4963  df-dir 15522
This theorem is referenced by:  reldir  15525  dirdm  15526  dirref  15527  dirtr  15528  dirge  15529  tsrdir  15530  filnetlem3  28769
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