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Theorem cnvun 5263
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 4872 . . 3  |-  `' ( A  u.  B )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
2 unopab 4271 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  \/  y B x ) }
3 brun 4245 . . . . 5  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
43opabbii 4259 . . . 4  |-  { <. x ,  y >.  |  y ( A  u.  B
) x }  =  { <. x ,  y
>.  |  ( y A x  \/  y B x ) }
52, 4eqtr4i 2453 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
61, 5eqtr4i 2453 . 2  |-  `' ( A  u.  B )  =  ( { <. x ,  y >.  |  y A x }  u.  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4872 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4872 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8uneq12i 3486 . 2  |-  ( `' A  u.  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2453 1  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1652    u. cun 3305   class class class wbr 4199   {copab 4252   `'ccnv 4863
This theorem is referenced by:  rnun  5266  f1oun  5680  f1oprswap  5703  sbthlem8  7210  domss2  7252  1sdom  7297  fpwwe2lem13  8501  strlemor1  13539  xpsc  13765  gsumzaddlem  15509  mbfres2  19520  constr2spthlem1  21577  constr3pthlem2  21626  ex-cnv  21728  funsnfsup  26675
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-un 3312  df-br 4200  df-opab 4254  df-cnv 4872
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