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Theorem cnvco 5048
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 1596 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 2951 . . . . 5  |-  x  e. 
_V
3 vex 2951 . . . . 5  |-  y  e. 
_V
42, 3brco 5035 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2951 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 5047 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 5047 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 679 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1592 . . . 4  |-  ( E. z ( y `' A z  /\  z `' B x )  <->  E. z
( z A y  /\  x B z ) )
101, 4, 93bitr4i 269 . . 3  |-  ( x ( A  o.  B
) y  <->  E. z
( y `' A
z  /\  z `' B x ) )
1110opabbii 4264 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4878 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4879 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2465 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652   class class class wbr 4204   {copab 4257   `'ccnv 4869    o. ccom 4874
This theorem is referenced by:  rncoss  5128  rncoeq  5131  dmco  5370  cores2  5374  co01  5376  coi2  5378  relcnvtr  5381  dfdm2  5393  f1co  5640  cofunex2g  5952  fparlem3  6440  fparlem4  6441  suppfif1  7392  mapfien  7645  cnvps  14636  gimco  15047  gsumval3  15506  gsumzf1o  15511  cnco  17322  ptrescn  17663  qtopcn  17738  hmeoco  17796  cncombf  19542  deg1val  20011  ofpreima  24073  mbfmco  24606  cvmliftmolem1  24960  cvmlift2lem9a  24982  cvmlift2lem9  24990  relexpcnv  25125  ftc1anclem3  26272  trlcocnv  31454  tendoicl  31530  cdlemk45  31681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cnv 4878  df-co 4879
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