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Theorem cnvco 4865
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 1573 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 2791 . . . . 5  |-  x  e. 
_V
3 vex 2791 . . . . 5  |-  y  e. 
_V
42, 3brco 4852 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2791 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 4864 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 4864 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 678 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1569 . . . 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 4083 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4697 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4698 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2313 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623   class class class wbr 4023   {copab 4076   `'ccnv 4688    o. ccom 4693
This theorem is referenced by:  rncoss  4945  rncoeq  4948  dmco  5181  cores2  5185  co01  5187  coi2  5189  relcnvtr  5192  dfdm2  5204  f1co  5446  cofunex2g  5740  fparlem3  6220  fparlem4  6221  suppfif1  7149  mapfien  7399  cnvps  14321  gimco  14732  gsumval3  15191  gsumzf1o  15196  cnco  16995  ptrescn  17333  qtopcn  17405  hmeoco  17463  cncombf  19013  deg1val  19482  mbfmco  23569  cvmliftmolem1  23812  cvmlift2lem9a  23834  cvmlift2lem9  23842  relexpcnv  24029  trlcocnv  30909  tendoicl  30985  cdlemk45  31136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-co 4698
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