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Theorem vtoclr 4594
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
21brrelex1i 4589 . . . . 5  |-  ( A R B  ->  A  e.  _V )
31brrelex2i 4590 . . . . 5  |-  ( A R B  ->  B  e.  _V )
42, 3jca 304 . . . 4  |-  ( A R B  ->  ( A  e.  _V  /\  B  e.  _V ) )
51brrelex2i 4590 . . . 4  |-  ( B R C  ->  C  e.  _V )
6 breq1 3939 . . . . . . . 8  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
76anbi1d 461 . . . . . . 7  |-  ( x  =  A  ->  (
( x R y  /\  y R C )  <->  ( A R y  /\  y R C ) ) )
8 breq1 3939 . . . . . . 7  |-  ( x  =  A  ->  (
x R C  <->  A R C ) )
97, 8imbi12d 233 . . . . . 6  |-  ( x  =  A  ->  (
( ( x R y  /\  y R C )  ->  x R C )  <->  ( ( A R y  /\  y R C )  ->  A R C ) ) )
109imbi2d 229 . . . . 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 3940 . . . . . . . 8  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
12 breq1 3939 . . . . . . . 8  |-  ( y  =  B  ->  (
y R C  <->  B R C ) )
1311, 12anbi12d 465 . . . . . . 7  |-  ( y  =  B  ->  (
( A R y  /\  y R C )  <->  ( A R B  /\  B R C ) ) )
1413imbi1d 230 . . . . . 6  |-  ( y  =  B  ->  (
( ( A R y  /\  y R C )  ->  A R C )  <->  ( ( A R B  /\  B R C )  ->  A R C ) ) )
1514imbi2d 229 . . . . 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 3940 . . . . . . . 8  |-  ( z  =  C  ->  (
y R z  <->  y R C ) )
1716anbi2d 460 . . . . . . 7  |-  ( z  =  C  ->  (
( x R y  /\  y R z )  <->  ( x R y  /\  y R C ) ) )
18 breq2 3940 . . . . . . 7  |-  ( z  =  C  ->  (
x R z  <->  x R C ) )
1917, 18imbi12d 233 . . . . . 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 2749 . . . . 5  |-  ( C  e.  _V  ->  (
( x R y  /\  y R C )  ->  x R C ) )
2210, 15, 21vtocl2g 2753 . . . 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 123 . 2  |-  ( ( A R B  /\  B R C )  -> 
( ( A R B  /\  B R C )  ->  A R C ) )
2524pm2.43i 49 1  |-  ( ( A R B  /\  B R C )  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   _Vcvv 2689   class class class wbr 3936   Rel wrel 4551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553
This theorem is referenced by:  domtr  6686
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