MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvun Unicode version

Theorem cnvun 5039
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
StepHypRef Expression
1 df-cnv 4642 . . 3  |-  `' ( A  u.  B )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
2 unopab 4035 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  \/  y B x ) }
3 brun 4009 . . . . 5  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
43opabbii 4023 . . . 4  |-  { <. x ,  y >.  |  y ( A  u.  B
) x }  =  { <. x ,  y
>.  |  ( y A x  \/  y B x ) }
52, 4eqtr4i 2279 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
61, 5eqtr4i 2279 . 2  |-  `' ( A  u.  B )  =  ( { <. x ,  y >.  |  y A x }  u.  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4642 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4642 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8uneq12i 3269 . 2  |-  ( `' A  u.  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2279 1  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
Colors of variables: wff set class
Syntax hints:    \/ wo 359    = wceq 1619    u. cun 3092   class class class wbr 3963   {copab 4016   `'ccnv 4625
This theorem is referenced by:  rnun  5042  f1oun  5395  f1oprswap  5418  sbthlem8  6911  domss2  6953  1sdom  6998  fpwwe2lem13  8197  strlemor1  13162  xpsc  13386  gsumzaddlem  15130  mbfres2  18927  ex-cnv  20732  funsnfsup  26094
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-un 3099  df-br 3964  df-opab 4018  df-cnv 4642
  Copyright terms: Public domain W3C validator