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Theorem cnvco 4773
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 1588 . . . 4  |-  ( E. z ( x B z  /\  z A y )  <->  E. z
( z A y  /\  x B z ) )
2 vex 2715 . . . . 5  |-  x  e. 
_V
3 vex 2715 . . . . 5  |-  y  e. 
_V
42, 3brco 4759 . . . 4  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
5 vex 2715 . . . . . . 7  |-  z  e. 
_V
63, 5brcnv 4771 . . . . . 6  |-  ( y `' A z  <->  z A
y )
75, 2brcnv 4771 . . . . . 6  |-  ( z `' B x  <->  x B
z )
86, 7anbi12i 456 . . . . 5  |-  ( ( y `' A z  /\  z `' B x )  <->  ( z A y  /\  x B z ) )
98exbii 1585 . . . 4  |-  ( E. z ( y `' A z  /\  z `' B x )  <->  E. z
( z A y  /\  x B z ) )
101, 4, 93bitr4i 211 . . 3  |-  ( x ( A  o.  B
) y  <->  E. z
( y `' A
z  /\  z `' B x ) )
1110opabbii 4033 . 2  |-  { <. y ,  x >.  |  x ( A  o.  B
) y }  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
12 df-cnv 4596 . 2  |-  `' ( A  o.  B )  =  { <. y ,  x >.  |  x
( A  o.  B
) y }
13 df-co 4597 . 2  |-  ( `' B  o.  `' A
)  =  { <. y ,  x >.  |  E. z ( y `' A z  /\  z `' B x ) }
1411, 12, 133eqtr4i 2188 1  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1335   E.wex 1472   class class class wbr 3967   {copab 4026   `'ccnv 4587    o. ccom 4592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-cnv 4596  df-co 4597
This theorem is referenced by:  rncoss  4858  rncoeq  4861  dmco  5096  cores2  5100  co01  5102  coi2  5104  relcnvtr  5107  dfdm2  5122  f1co  5389  cofunex2g  6062  caseinj  7035  djuinj  7052  cnco  12691  hmeoco  12786
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