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Theorem cnvco 5056
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 2959 . . . . 5  |-  x  e. 
_V
3 vex 2959 . . . . 5  |-  y  e. 
_V
42, 3brco 5043 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2959 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 5055 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 5055 . . . . . 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 4272 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4886 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4887 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2466 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 4212   {copab 4265   `'ccnv 4877    o. ccom 4882
This theorem is referenced by:  rncoss  5136  rncoeq  5139  dmco  5378  cores2  5382  co01  5384  coi2  5386  relcnvtr  5389  dfdm2  5401  f1co  5648  cofunex2g  5960  fparlem3  6448  fparlem4  6449  suppfif1  7400  mapfien  7653  cnvps  14644  gimco  15055  gsumval3  15514  gsumzf1o  15519  cnco  17330  ptrescn  17671  qtopcn  17746  hmeoco  17804  cncombf  19550  deg1val  20019  ofpreima  24081  mbfmco  24614  cvmliftmolem1  24968  cvmlift2lem9a  24990  cvmlift2lem9  24998  relexpcnv  25133  ftc1anclem3  26282  trlcocnv  31517  tendoicl  31593  cdlemk45  31744
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-co 4887
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