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Theorem cnvun 5280
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 4889 . . 3  |-  `' ( A  u.  B )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
2 unopab 4287 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  \/  y B x ) }
3 brun 4261 . . . . 5  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
43opabbii 4275 . . . 4  |-  { <. x ,  y >.  |  y ( A  u.  B
) x }  =  { <. x ,  y
>.  |  ( y A x  \/  y B x ) }
52, 4eqtr4i 2461 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  u.  B
) x }
61, 5eqtr4i 2461 . 2  |-  `' ( A  u.  B )  =  ( { <. x ,  y >.  |  y A x }  u.  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4889 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4889 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8uneq12i 3501 . 2  |-  ( `' A  u.  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  u.  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2461 1  |-  `' ( A  u.  B )  =  ( `' A  u.  `' B )
Colors of variables: wff set class
Syntax hints:    \/ wo 359    = wceq 1653    u. cun 3320   class class class wbr 4215   {copab 4268   `'ccnv 4880
This theorem is referenced by:  rnun  5283  f1oun  5697  f1oprswap  5720  sbthlem8  7227  domss2  7269  1sdom  7314  fpwwe2lem13  8522  strlemor1  13561  xpsc  13787  gsumzaddlem  15531  mbfres2  19540  constr2spthlem1  21599  constr3pthlem2  21648  ex-cnv  21750  funsnfsup  26757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-br 4216  df-opab 4270  df-cnv 4889
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