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Theorem vtoclr 4895
Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
vtoclr.1  |-  Rel  R
vtoclr.2  |-  ( ( x R y  /\  y R z )  ->  x R z )
Assertion
Ref Expression
vtoclr  |-  ( ( A R B  /\  B R C )  ->  A R C )
Distinct variable groups:    x, y, A    y, B    x, z, C, y    x, R, y, z
Allowed substitution hints:    A( z)    B( x, z)

Proof of Theorem vtoclr
StepHypRef Expression
1 vtoclr.1 . . . . . 6  |-  Rel  R
21brrelexi 4891 . . . . 5  |-  ( A R B  ->  A  e.  _V )
31brrelex2i 4892 . . . . 5  |-  ( A R B  ->  B  e.  _V )
42, 3jca 532 . . . 4  |-  ( A R B  ->  ( A  e.  _V  /\  B  e.  _V ) )
51brrelex2i 4892 . . . 4  |-  ( B R C  ->  C  e.  _V )
6 breq1 4307 . . . . . . . 8  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
76anbi1d 704 . . . . . . 7  |-  ( x  =  A  ->  (
( x R y  /\  y R C )  <->  ( A R y  /\  y R C ) ) )
8 breq1 4307 . . . . . . 7  |-  ( x  =  A  ->  (
x R C  <->  A R C ) )
97, 8imbi12d 320 . . . . . 6  |-  ( x  =  A  ->  (
( ( x R y  /\  y R C )  ->  x R C )  <->  ( ( A R y  /\  y R C )  ->  A R C ) ) )
109imbi2d 316 . . . . 5  |-  ( x  =  A  ->  (
( C  e.  _V  ->  ( ( x R y  /\  y R C )  ->  x R C ) )  <->  ( C  e.  _V  ->  ( ( A R y  /\  y R C )  ->  A R C ) ) ) )
11 breq2 4308 . . . . . . . 8  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
12 breq1 4307 . . . . . . . 8  |-  ( y  =  B  ->  (
y R C  <->  B R C ) )
1311, 12anbi12d 710 . . . . . . 7  |-  ( y  =  B  ->  (
( A R y  /\  y R C )  <->  ( A R B  /\  B R C ) ) )
1413imbi1d 317 . . . . . 6  |-  ( y  =  B  ->  (
( ( A R y  /\  y R C )  ->  A R C )  <->  ( ( A R B  /\  B R C )  ->  A R C ) ) )
1514imbi2d 316 . . . . 5  |-  ( y  =  B  ->  (
( C  e.  _V  ->  ( ( A R y  /\  y R C )  ->  A R C ) )  <->  ( C  e.  _V  ->  ( ( A R B  /\  B R C )  ->  A R C ) ) ) )
16 breq2 4308 . . . . . . . 8  |-  ( z  =  C  ->  (
y R z  <->  y R C ) )
1716anbi2d 703 . . . . . . 7  |-  ( z  =  C  ->  (
( x R y  /\  y R z )  <->  ( x R y  /\  y R C ) ) )
18 breq2 4308 . . . . . . 7  |-  ( z  =  C  ->  (
x R z  <->  x R C ) )
1917, 18imbi12d 320 . . . . . 6  |-  ( z  =  C  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( (
x R y  /\  y R C )  ->  x R C ) ) )
20 vtoclr.2 . . . . . 6  |-  ( ( x R y  /\  y R z )  ->  x R z )
2119, 20vtoclg 3042 . . . . 5  |-  ( C  e.  _V  ->  (
( x R y  /\  y R C )  ->  x R C ) )
2210, 15, 21vtocl2g 3046 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( C  e.  _V  ->  ( ( A R B  /\  B R C )  ->  A R C ) ) )
234, 5, 22syl2im 38 . . 3  |-  ( A R B  ->  ( B R C  ->  (
( A R B  /\  B R C )  ->  A R C ) ) )
2423imp 429 . 2  |-  ( ( A R B  /\  B R C )  -> 
( ( A R B  /\  B R C )  ->  A R C ) )
2524pm2.43i 47 1  |-  ( ( A R B  /\  B R C )  ->  A R C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2984   class class class wbr 4304   Rel wrel 4857
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 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
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-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-xp 4858  df-rel 4859
This theorem is referenced by:  domtr  7374
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