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Theorem cnvco 4881
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 1576 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 2804 . . . . 5  |-  x  e. 
_V
3 vex 2804 . . . . 5  |-  y  e. 
_V
42, 3brco 4868 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2804 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 4880 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 4880 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 678 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1572 . . . 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 4099 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4713 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4714 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2326 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632   class class class wbr 4039   {copab 4092   `'ccnv 4704    o. ccom 4709
This theorem is referenced by:  rncoss  4961  rncoeq  4964  dmco  5197  cores2  5201  co01  5203  coi2  5205  relcnvtr  5208  dfdm2  5220  f1co  5462  cofunex2g  5756  fparlem3  6236  fparlem4  6237  suppfif1  7165  mapfien  7415  cnvps  14337  gimco  14748  gsumval3  15207  gsumzf1o  15212  cnco  17011  ptrescn  17349  qtopcn  17421  hmeoco  17479  cncombf  19029  deg1val  19498  mbfmco  23584  cvmliftmolem1  23827  cvmlift2lem9a  23849  cvmlift2lem9  23857  relexpcnv  24044  trlcocnv  31531  tendoicl  31607  cdlemk45  31758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-co 4714
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