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Theorem dmun 4838
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
StepHypRef Expression
1 unab 3377 . . 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 4009 . . . . . 6  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
32exbii 1580 . . . . 5  |-  ( E. x  y ( A  u.  B ) x  <->  E. x ( y A x  \/  y B x ) )
4 19.43 1604 . . . . 5  |-  ( E. x ( y A x  \/  y B x )  <->  ( E. x  y A x  \/  E. x  y B x ) )
53, 4bitr2i 243 . . . 4  |-  ( ( E. x  y A x  \/  E. x  y B x )  <->  E. x  y ( A  u.  B ) x )
65abbii 2368 . . 3  |-  { y  |  ( E. x  y A x  \/  E. x  y B x ) }  =  {
y  |  E. x  y ( A  u.  B ) x }
71, 6eqtri 2276 . 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 4644 . . 3  |-  dom  A  =  { y  |  E. x  y A x }
9 df-dm 4644 . . 3  |-  dom  B  =  { y  |  E. x  y B x }
108, 9uneq12i 3269 . 2  |-  ( dom 
A  u.  dom  B
)  =  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )
11 df-dm 4644 . 2  |-  dom  (  A  u.  B )  =  { y  |  E. x  y ( A  u.  B ) x }
127, 10, 113eqtr4ri 2287 1  |-  dom  (  A  u.  B )  =  ( dom  A  u.  dom  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 359   E.wex 1537    = wceq 1619   {cab 2242    u. cun 3092   class class class wbr 3963   dom cdm 4626
This theorem is referenced by:  rnun  5042  dmpropg  5098  dmtpop  5101  fnun  5253  tfrlem10  6336  sbthlem5  6908  fodomr  6945  axdc3lem4  8012  hashfun  11319  strlemor1  13162  strleun  13165  xpsfrnel2  13394  wfrlem13  23602  wfrlem16  23605  fixun  23790  mvdco  26720  bnj1416  28081
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-dm 4644
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