MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isdir Structured version   Unicode version

Theorem isdir 15736
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 5091 . . . 4  |-  ( r  =  R  ->  ( Rel  r  <->  Rel  R ) )
2 unieq 4259 . . . . . . . 8  |-  ( r  =  R  ->  U. r  =  U. R )
32unieqd 4261 . . . . . . 7  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
4 isdir.1 . . . . . . 7  |-  A  = 
U. U. R
53, 4syl6eqr 2526 . . . . . 6  |-  ( r  =  R  ->  U. U. r  =  A )
65reseq2d 5279 . . . . 5  |-  ( r  =  R  ->  (  _I  |`  U. U. r
)  =  (  _I  |`  A ) )
7 id 22 . . . . 5  |-  ( r  =  R  ->  r  =  R )
86, 7sseq12d 3538 . . . 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 5173 . . . . 5  |-  ( r  =  R  ->  (
r  o.  r )  =  ( R  o.  R ) )
1110, 7sseq12d 3538 . . . 4  |-  ( r  =  R  ->  (
( r  o.  r
)  C_  r  <->  ( R  o.  R )  C_  R
) )
125sqxpeqd 5031 . . . . 5  |-  ( r  =  R  ->  ( U. U. r  X.  U. U. r )  =  ( A  X.  A ) )
13 cnveq 5182 . . . . . 6  |-  ( r  =  R  ->  `' r  =  `' R
)
1413, 7coeq12d 5173 . . . . 5  |-  ( r  =  R  ->  ( `' r  o.  r
)  =  ( `' R  o.  R ) )
1512, 14sseq12d 3538 . . . 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 15734 . 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 3257 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 1379    e. wcel 1767    C_ wss 3481   U.cuni 4251    _I cid 4796    X. cxp 5003   `'ccnv 5004    |` cres 5007    o. ccom 5009   Rel wrel 5010   DirRelcdir 15732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-v 3120  df-in 3488  df-ss 3495  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-res 5017  df-dir 15734
This theorem is referenced by:  reldir  15737  dirdm  15738  dirref  15739  dirtr  15740  dirge  15741  tsrdir  15742  filnetlem3  30125
  Copyright terms: Public domain W3C validator