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Theorem dirtr 15404
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 15401 . . . . 5  |-  ( R  e.  DirRel  ->  Rel  R )
2 brrelex 4875 . . . . . . 7  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
32ex 434 . . . . . 6  |-  ( Rel 
R  ->  ( A R B  ->  A  e. 
_V ) )
4 brrelex 4875 . . . . . . 7  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
54ex 434 . . . . . 6  |-  ( Rel 
R  ->  ( B R C  ->  B  e. 
_V ) )
63, 5anim12d 563 . . . . 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 2441 . . . . . . . . . . . 12  |-  U. U. R  =  U. U. R
98isdir 15400 . . . . . . . . . . 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 241 . . . . . . . . . 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 463 . . . . . . . . 9  |-  ( R  e.  DirRel  ->  ( ( R  o.  R )  C_  R  /\  ( U. U. R  X.  U. U. R
)  C_  ( `' R  o.  R )
) )
1211simpld 459 . . . . . . . 8  |-  ( R  e.  DirRel  ->  ( R  o.  R )  C_  R
)
13 cotr 5208 . . . . . . . 8  |-  ( ( R  o.  R ) 
C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R
z ) )
1412, 13sylib 196 . . . . . . 7  |-  ( R  e.  DirRel  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
15 breq12 4295 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x R y  <-> 
A R B ) )
16153adant3 1008 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R y  <-> 
A R B ) )
17 breq12 4295 . . . . . . . . . . 11  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
18173adant1 1006 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
1916, 18anbi12d 710 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x R y  /\  y R z )  <->  ( A R B  /\  B R C ) ) )
20 breq12 4295 . . . . . . . . . 10  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
21203adant2 1007 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
2219, 21imbi12d 320 . . . . . . . 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 3060 . . . . . . 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 32 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  V )  ->  ( R  e.  DirRel  ->  (
( A R B  /\  B R C )  ->  A R C ) ) )
25243expia 1189 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( C  e.  V  ->  ( R  e.  DirRel  -> 
( ( A R B  /\  B R C )  ->  A R C ) ) ) )
2625com4t 85 . . . 4  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( C  e.  V  ->  A R C ) ) ) )
277, 26mpdd 40 . . 3  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  ( C  e.  V  ->  A R C ) ) )
2827imp31 432 . 2  |-  ( ( ( R  e.  DirRel  /\  ( A R B  /\  B R C ) )  /\  C  e.  V )  ->  A R C )
2928an32s 802 1  |-  ( ( ( R  e.  DirRel  /\  C  e.  V )  /\  ( A R B  /\  B R C ) )  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   _Vcvv 2970    C_ wss 3326   U.cuni 4089   class class class wbr 4290    _I cid 4629    X. cxp 4836   `'ccnv 4837    |` cres 4840    o. ccom 4842   Rel wrel 4843   DirRelcdir 15396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-res 4850  df-dir 15398
This theorem is referenced by:  tailfb  28595
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