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Theorem dmun 5207
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 3765 . . 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 4495 . . . . . 6  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
32exbii 1644 . . . . 5  |-  ( E. x  y ( A  u.  B ) x  <->  E. x ( y A x  \/  y B x ) )
4 19.43 1670 . . . . 5  |-  ( E. x ( y A x  \/  y B x )  <->  ( E. x  y A x  \/  E. x  y B x ) )
53, 4bitr2i 250 . . . 4  |-  ( ( E. x  y A x  \/  E. x  y B x )  <->  E. x  y ( A  u.  B ) x )
65abbii 2601 . . 3  |-  { y  |  ( E. x  y A x  \/  E. x  y B x ) }  =  {
y  |  E. x  y ( A  u.  B ) x }
71, 6eqtri 2496 . 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 5009 . . 3  |-  dom  A  =  { y  |  E. x  y A x }
9 df-dm 5009 . . 3  |-  dom  B  =  { y  |  E. x  y B x }
108, 9uneq12i 3656 . 2  |-  ( dom 
A  u.  dom  B
)  =  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )
11 df-dm 5009 . 2  |-  dom  ( A  u.  B )  =  { y  |  E. x  y ( A  u.  B ) x }
127, 10, 113eqtr4ri 2507 1  |-  dom  ( A  u.  B )  =  ( dom  A  u.  dom  B )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1379   E.wex 1596   {cab 2452    u. cun 3474   class class class wbr 4447   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-br 4448  df-dm 5009
This theorem is referenced by:  rnun  5412  dmpropg  5479  dmtpop  5482  fntpg  5641  fnun  5685  tfrlem10  7053  sbthlem5  7628  fodomr  7665  axdc3lem4  8829  hashfun  12457  s4dom  12826  strlemor1  14578  strleun  14581  xpsfrnel2  14816  mvdco  16266  gsumzaddlem  16725  wfrlem13  28932  wfrlem16  28935  fixun  29136  bnj1416  33174
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