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Theorem dirtr 14673
Description: A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Assertion
Ref Expression
dirtr  |-  ( ( ( R  e.  DirRel  /\  C  e.  V )  /\  ( A R B  /\  B R C ) )  ->  A R C )

Proof of Theorem dirtr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldir 14670 . . . . 5  |-  ( R  e.  DirRel  ->  Rel  R )
2 brrelex 4908 . . . . . . 7  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
32ex 424 . . . . . 6  |-  ( Rel 
R  ->  ( A R B  ->  A  e. 
_V ) )
4 brrelex 4908 . . . . . . 7  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
54ex 424 . . . . . 6  |-  ( Rel 
R  ->  ( B R C  ->  B  e. 
_V ) )
63, 5anim12d 547 . . . . 5  |-  ( Rel 
R  ->  ( ( A R B  /\  B R C )  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
71, 6syl 16 . . . 4  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
8 eqid 2435 . . . . . . . . . . . 12  |-  U. U. R  =  U. U. R
98isdir 14669 . . . . . . . . . . 11  |-  ( 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
) ) ) ) )
109ibi 233 . . . . . . . . . 10  |-  ( 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
) ) ) )
1110simprd 450 . . . . . . . . 9  |-  ( R  e.  DirRel  ->  ( ( R  o.  R )  C_  R  /\  ( U. U. R  X.  U. U. R
)  C_  ( `' R  o.  R )
) )
1211simpld 446 . . . . . . . 8  |-  ( R  e.  DirRel  ->  ( R  o.  R )  C_  R
)
13 cotr 5238 . . . . . . . 8  |-  ( ( R  o.  R ) 
C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R
z ) )
1412, 13sylib 189 . . . . . . 7  |-  ( R  e.  DirRel  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
15 breq12 4209 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x R y  <-> 
A R B ) )
16153adant3 977 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R y  <-> 
A R B ) )
17 breq12 4209 . . . . . . . . . . 11  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
18173adant1 975 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
1916, 18anbi12d 692 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x R y  /\  y R z )  <->  ( A R B  /\  B R C ) ) )
20 breq12 4209 . . . . . . . . . 10  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
21203adant2 976 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
2219, 21imbi12d 312 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x R y  /\  y R z )  ->  x R z )  <->  ( ( A R B  /\  B R C )  ->  A R C ) ) )
2322spc3gv 3033 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  V )  ->  ( A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z )  -> 
( ( A R B  /\  B R C )  ->  A R C ) ) )
2414, 23syl5 30 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  V )  ->  ( R  e.  DirRel  ->  (
( A R B  /\  B R C )  ->  A R C ) ) )
25243expia 1155 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( C  e.  V  ->  ( R  e.  DirRel  -> 
( ( A R B  /\  B R C )  ->  A R C ) ) ) )
2625com4t 81 . . . 4  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( C  e.  V  ->  A R C ) ) ) )
277, 26mpdd 38 . . 3  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  ( C  e.  V  ->  A R C ) ) )
2827imp31 422 . 2  |-  ( ( ( R  e.  DirRel  /\  ( A R B  /\  B R C ) )  /\  C  e.  V )  ->  A R C )
2928an32s 780 1  |-  ( ( ( R  e.  DirRel  /\  C  e.  V )  /\  ( A R B  /\  B R C ) )  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   U.cuni 4007   class class class wbr 4204    _I cid 4485    X. cxp 4868   `'ccnv 4869    |` cres 4872    o. ccom 4874   Rel wrel 4875   DirRelcdir 14665
This theorem is referenced by:  tailfb  26397
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-res 4882  df-dir 14667
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