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Theorem dmun 4901
Description: The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmun  |-  dom  ( A  u.  B )  =  ( dom  A  u.  dom  B )

Proof of Theorem dmun
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unab 3448 . . 3  |-  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )  =  { y  |  ( E. x  y A x  \/  E. x  y B x ) }
2 brun 4085 . . . . . 6  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
32exbii 1572 . . . . 5  |-  ( E. x  y ( A  u.  B ) x  <->  E. x ( y A x  \/  y B x ) )
4 19.43 1595 . . . . 5  |-  ( E. x ( y A x  \/  y B x )  <->  ( E. x  y A x  \/  E. x  y B x ) )
53, 4bitr2i 241 . . . 4  |-  ( ( E. x  y A x  \/  E. x  y B x )  <->  E. x  y ( A  u.  B ) x )
65abbii 2408 . . 3  |-  { y  |  ( E. x  y A x  \/  E. x  y B x ) }  =  {
y  |  E. x  y ( A  u.  B ) x }
71, 6eqtri 2316 . 2  |-  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )  =  { y  |  E. x  y ( A  u.  B ) x }
8 df-dm 4715 . . 3  |-  dom  A  =  { y  |  E. x  y A x }
9 df-dm 4715 . . 3  |-  dom  B  =  { y  |  E. x  y B x }
108, 9uneq12i 3340 . 2  |-  ( dom 
A  u.  dom  B
)  =  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )
11 df-dm 4715 . 2  |-  dom  ( A  u.  B )  =  { y  |  E. x  y ( A  u.  B ) x }
127, 10, 113eqtr4ri 2327 1  |-  dom  ( A  u.  B )  =  ( dom  A  u.  dom  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 357   E.wex 1531    = wceq 1632   {cab 2282    u. cun 3163   class class class wbr 4039   dom cdm 4705
This theorem is referenced by:  rnun  5105  dmpropg  5162  dmtpop  5165  fnun  5366  tfrlem10  6419  sbthlem5  6991  fodomr  7028  axdc3lem4  8095  hashfun  11405  strlemor1  13251  strleun  13254  xpsfrnel2  13483  wfrlem13  24339  wfrlem16  24342  fixun  24520  mvdco  27491  s4dom  28224  bnj1416  29385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-br 4040  df-dm 4715
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