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Theorem cnvun 5052
Description: The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvun  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )

Proof of Theorem cnvun
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4652 . . 3  |-  `' ( A  u.  B )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
2 unopab 4097 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  \/  y B x ) }
3 brun 4069 . . . . 5  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
43opabbii 4085 . . . 4  |-  { <. x ,  y >.  |  y ( A  u.  B
) x }  =  { <. x ,  y
>.  |  ( y A x  \/  y B x ) }
52, 4eqtr4i 2213 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
61, 5eqtr4i 2213 . 2  |-  `' ( A  u.  B )  =  ( { <. x ,  y >.  |  y A x }  u.  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4652 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4652 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8uneq12i 3302 . 2  |-  ( `' A  u.  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2213 1  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
Colors of variables: wff set class
Syntax hints:    \/ wo 709    = wceq 1364    u. cun 3142   class class class wbr 4018   {copab 4078   `'ccnv 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-br 4019  df-opab 4080  df-cnv 4652
This theorem is referenced by:  rnun  5055  f1oun  5500  sbthlemi8  6993  caseinj  7118  djuinj  7135
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