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Theorem cnvco 5194
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 1648 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 3121 . . . . 5  |-  x  e. 
_V
3 vex 3121 . . . . 5  |-  y  e. 
_V
42, 3brco 5179 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 3121 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 5191 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 5191 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 697 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1644 . . . 4  |-  ( E. z ( y `' A z  /\  z `' B x )  <->  E. z
( z A y  /\  x B z ) )
101, 4, 93bitr4i 277 . . 3  |-  ( x ( A  o.  B
) y  <->  E. z
( y `' A
z  /\  z `' B x ) )
1110opabbii 4517 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 5013 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 5014 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2506 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596   class class class wbr 4453   {copab 4510   `'ccnv 5004    o. ccom 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-cnv 5013  df-co 5014
This theorem is referenced by:  rncoss  5269  rncoeq  5272  dmco  5521  cores2  5526  co01  5528  coi2  5530  relcnvtr  5533  dfdm2  5545  f1co  5796  cofunex2g  6760  fparlem3  6897  fparlem4  6898  supp0cosupp0  6951  imacosupp  6952  fsuppcolem  7872  mapfienOLD  8150  cnvps  15716  gimco  16188  gsumval3OLD  16781  gsumzf1o  16790  gsumzf1oOLD  16793  cnco  19635  ptrescn  20008  qtopcn  20083  hmeoco  20141  cncombf  21933  deg1val  22364  deg1valOLD  22365  fcoinver  27274  ofpreima  27321  mbfmco  28051  eulerpartlemmf  28130  cvmliftmolem1  28542  cvmlift2lem9a  28564  cvmlift2lem9  28572  mclsppslem  28759  relexpcnv  28872  ftc1anclem3  30010  trlcocnv  35872  tendoicl  35948  cdlemk45  36099  cnvtrrel  37148
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