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Theorem dirtr 16418
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 16415 . . . . 5  |-  ( R  e.  DirRel  ->  Rel  R )
2 brrelex 4828 . . . . . . 7  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
32ex 435 . . . . . 6  |-  ( Rel 
R  ->  ( A R B  ->  A  e. 
_V ) )
4 brrelex 4828 . . . . . . 7  |-  ( ( Rel  R  /\  B R C )  ->  B  e.  _V )
54ex 435 . . . . . 6  |-  ( Rel 
R  ->  ( B R C  ->  B  e. 
_V ) )
63, 5anim12d 565 . . . . 5  |-  ( Rel 
R  ->  ( ( A R B  /\  B R C )  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
71, 6syl 17 . . . 4  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  ( A  e.  _V  /\  B  e.  _V ) ) )
8 eqid 2422 . . . . . . . . . . . 12  |-  U. U. R  =  U. U. R
98isdir 16414 . . . . . . . . . . 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 244 . . . . . . . . . 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 464 . . . . . . . . 9  |-  ( R  e.  DirRel  ->  ( ( R  o.  R )  C_  R  /\  ( U. U. R  X.  U. U. R
)  C_  ( `' R  o.  R )
) )
1211simpld 460 . . . . . . . 8  |-  ( R  e.  DirRel  ->  ( R  o.  R )  C_  R
)
13 cotr 5167 . . . . . . . 8  |-  ( ( R  o.  R ) 
C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R
z ) )
1412, 13sylib 199 . . . . . . 7  |-  ( R  e.  DirRel  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
15 breq12 4364 . . . . . . . . . . 11  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x R y  <-> 
A R B ) )
16153adant3 1025 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R y  <-> 
A R B ) )
17 breq12 4364 . . . . . . . . . . 11  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
18173adant1 1023 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
1916, 18anbi12d 715 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x R y  /\  y R z )  <->  ( A R B  /\  B R C ) ) )
20 breq12 4364 . . . . . . . . . 10  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
21203adant2 1024 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
2219, 21imbi12d 321 . . . . . . . 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 3107 . . . . . . 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 33 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  V )  ->  ( R  e.  DirRel  ->  (
( A R B  /\  B R C )  ->  A R C ) ) )
25243expia 1207 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( C  e.  V  ->  ( R  e.  DirRel  -> 
( ( A R B  /\  B R C )  ->  A R C ) ) ) )
2625com4t 88 . . . 4  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  (
( A  e.  _V  /\  B  e.  _V )  ->  ( C  e.  V  ->  A R C ) ) ) )
277, 26mpdd 41 . . 3  |-  ( R  e.  DirRel  ->  ( ( A R B  /\  B R C )  ->  ( C  e.  V  ->  A R C ) ) )
2827imp31 433 . 2  |-  ( ( ( R  e.  DirRel  /\  ( A R B  /\  B R C ) )  /\  C  e.  V )  ->  A R C )
2928an32s 811 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 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1872   _Vcvv 3016    C_ wss 3372   U.cuni 4155   class class class wbr 4359    _I cid 4699    X. cxp 4787   `'ccnv 4788    |` cres 4791    o. ccom 4793   Rel wrel 4794   DirRelcdir 16410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pr 4596
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-res 4801  df-dir 16412
This theorem is referenced by:  tailfb  30975
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