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Theorem cnvco 4968
Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvco  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )

Proof of Theorem cnvco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exancom 1591 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 2876 . . . . 5  |-  x  e. 
_V
3 vex 2876 . . . . 5  |-  y  e. 
_V
42, 3brco 4955 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2876 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 4967 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 4967 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 678 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1587 . . . 4  |-  ( E. z ( y `' A z  /\  z `' B x )  <->  E. z
( z A y  /\  x B z ) )
101, 4, 93bitr4i 268 . . 3  |-  ( x ( A  o.  B
) y  <->  E. z
( y `' A
z  /\  z `' B x ) )
1110opabbii 4185 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4800 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4801 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2396 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1546    = wceq 1647   class class class wbr 4125   {copab 4178   `'ccnv 4791    o. ccom 4796
This theorem is referenced by:  rncoss  5048  rncoeq  5051  dmco  5284  cores2  5288  co01  5290  coi2  5292  relcnvtr  5295  dfdm2  5307  f1co  5552  cofunex2g  5860  fparlem3  6348  fparlem4  6349  suppfif1  7296  mapfien  7546  cnvps  14531  gimco  14942  gsumval3  15401  gsumzf1o  15406  cnco  17212  ptrescn  17550  qtopcn  17622  hmeoco  17680  cncombf  19228  deg1val  19697  mbfmco  24077  cvmliftmolem1  24415  cvmlift2lem9a  24437  cvmlift2lem9  24445  relexpcnv  24616  trlcocnv  30980  tendoicl  31056  cdlemk45  31207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-cnv 4800  df-co 4801
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