MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caovord Unicode version

Theorem caovord 6197
Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)
Hypotheses
Ref Expression
caovord.1  |-  A  e. 
_V
caovord.2  |-  B  e. 
_V
caovord.3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
Assertion
Ref Expression
caovord  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, F, y, z    x, R, y, z    x, S, y, z

Proof of Theorem caovord
StepHypRef Expression
1 oveq1 6027 . . . 4  |-  ( z  =  C  ->  (
z F A )  =  ( C F A ) )
2 oveq1 6027 . . . 4  |-  ( z  =  C  ->  (
z F B )  =  ( C F B ) )
31, 2breq12d 4166 . . 3  |-  ( z  =  C  ->  (
( z F A ) R ( z F B )  <->  ( C F A ) R ( C F B ) ) )
43bibi2d 310 . 2  |-  ( z  =  C  ->  (
( A R B  <-> 
( z F A ) R ( z F B ) )  <-> 
( A R B  <-> 
( C F A ) R ( C F B ) ) ) )
5 caovord.1 . . 3  |-  A  e. 
_V
6 caovord.2 . . 3  |-  B  e. 
_V
7 breq1 4156 . . . . . 6  |-  ( x  =  A  ->  (
x R y  <->  A R
y ) )
8 oveq2 6028 . . . . . . 7  |-  ( x  =  A  ->  (
z F x )  =  ( z F A ) )
98breq1d 4163 . . . . . 6  |-  ( x  =  A  ->  (
( z F x ) R ( z F y )  <->  ( z F A ) R ( z F y ) ) )
107, 9bibi12d 313 . . . . 5  |-  ( x  =  A  ->  (
( x R y  <-> 
( z F x ) R ( z F y ) )  <-> 
( A R y  <-> 
( z F A ) R ( z F y ) ) ) )
11 breq2 4157 . . . . . 6  |-  ( y  =  B  ->  ( A R y  <->  A R B ) )
12 oveq2 6028 . . . . . . 7  |-  ( y  =  B  ->  (
z F y )  =  ( z F B ) )
1312breq2d 4165 . . . . . 6  |-  ( y  =  B  ->  (
( z F A ) R ( z F y )  <->  ( z F A ) R ( z F B ) ) )
1411, 13bibi12d 313 . . . . 5  |-  ( y  =  B  ->  (
( A R y  <-> 
( z F A ) R ( z F y ) )  <-> 
( A R B  <-> 
( z F A ) R ( z F B ) ) ) )
1510, 14sylan9bb 681 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x R y  <->  ( z F x ) R ( z F y ) )  <->  ( A R B  <->  ( z F A ) R ( z F B ) ) ) )
1615imbi2d 308 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( z  e.  S  ->  ( x R y  <->  ( z F x ) R ( z F y ) ) )  <->  ( z  e.  S  ->  ( A R B  <->  ( z F A ) R ( z F B ) ) ) ) )
17 caovord.3 . . 3  |-  ( z  e.  S  ->  (
x R y  <->  ( z F x ) R ( z F y ) ) )
185, 6, 16, 17vtocl2 2950 . 2  |-  ( z  e.  S  ->  ( A R B  <->  ( z F A ) R ( z F B ) ) )
194, 18vtoclga 2960 1  |-  ( C  e.  S  ->  ( A R B  <->  ( C F A ) R ( C F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   class class class wbr 4153  (class class class)co 6020
This theorem is referenced by:  caovord2  6198  caovord3  6199  genpcl  8818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023
  Copyright terms: Public domain W3C validator