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Theorem cnvco 5019
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 1593 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 2923 . . . . 5  |-  x  e. 
_V
3 vex 2923 . . . . 5  |-  y  e. 
_V
42, 3brco 5006 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2923 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 5018 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 5018 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 679 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1589 . . . 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 4236 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4849 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4850 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2438 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649   class class class wbr 4176   {copab 4229   `'ccnv 4840    o. ccom 4845
This theorem is referenced by:  rncoss  5099  rncoeq  5102  dmco  5341  cores2  5345  co01  5347  coi2  5349  relcnvtr  5352  dfdm2  5364  f1co  5611  cofunex2g  5923  fparlem3  6411  fparlem4  6412  suppfif1  7362  mapfien  7613  cnvps  14603  gimco  15014  gsumval3  15473  gsumzf1o  15478  cnco  17288  ptrescn  17628  qtopcn  17703  hmeoco  17761  cncombf  19507  deg1val  19976  ofpreima  24038  mbfmco  24571  cvmliftmolem1  24925  cvmlift2lem9a  24947  cvmlift2lem9  24955  relexpcnv  25090  trlcocnv  31206  tendoicl  31282  cdlemk45  31433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-cnv 4849  df-co 4850
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