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Theorem dmun 5043
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 3576 . . 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 4226 . . . . . 6  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
32exbii 1589 . . . . 5  |-  ( E. x  y ( A  u.  B ) x  <->  E. x ( y A x  \/  y B x ) )
4 19.43 1612 . . . . 5  |-  ( E. x ( y A x  \/  y B x )  <->  ( E. x  y A x  \/  E. x  y B x ) )
53, 4bitr2i 242 . . . 4  |-  ( ( E. x  y A x  \/  E. x  y B x )  <->  E. x  y ( A  u.  B ) x )
65abbii 2524 . . 3  |-  { y  |  ( E. x  y A x  \/  E. x  y B x ) }  =  {
y  |  E. x  y ( A  u.  B ) x }
71, 6eqtri 2432 . 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 4855 . . 3  |-  dom  A  =  { y  |  E. x  y A x }
9 df-dm 4855 . . 3  |-  dom  B  =  { y  |  E. x  y B x }
108, 9uneq12i 3467 . 2  |-  ( dom 
A  u.  dom  B
)  =  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )
11 df-dm 4855 . 2  |-  dom  ( A  u.  B )  =  { y  |  E. x  y ( A  u.  B ) x }
127, 10, 113eqtr4ri 2443 1  |-  dom  ( A  u.  B )  =  ( dom  A  u.  dom  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 358   E.wex 1547    = wceq 1649   {cab 2398    u. cun 3286   class class class wbr 4180   dom cdm 4845
This theorem is referenced by:  rnun  5247  dmpropg  5310  dmtpop  5313  fntpg  5473  fnun  5518  tfrlem10  6615  sbthlem5  7188  fodomr  7225  axdc3lem4  8297  hashfun  11663  s4dom  11829  strlemor1  13519  strleun  13522  xpsfrnel2  13753  wfrlem13  25490  wfrlem16  25493  fixun  25671  mvdco  27264  bnj1416  29126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-un 3293  df-br 4181  df-dm 4855
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