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Theorem dmun 4818
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 3394 . . 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 4040 . . . . . 6  |-  ( y ( A  u.  B
) x  <->  ( y A x  \/  y B x ) )
32exbii 1598 . . . . 5  |-  ( E. x  y ( A  u.  B ) x  <->  E. x ( y A x  \/  y B x ) )
4 19.43 1621 . . . . 5  |-  ( E. x ( y A x  \/  y B x )  <->  ( E. x  y A x  \/  E. x  y B x ) )
53, 4bitr2i 184 . . . 4  |-  ( ( E. x  y A x  \/  E. x  y B x )  <->  E. x  y ( A  u.  B ) x )
65abbii 2286 . . 3  |-  { y  |  ( E. x  y A x  \/  E. x  y B x ) }  =  {
y  |  E. x  y ( A  u.  B ) x }
71, 6eqtri 2191 . 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 4621 . . 3  |-  dom  A  =  { y  |  E. x  y A x }
9 df-dm 4621 . . 3  |-  dom  B  =  { y  |  E. x  y B x }
108, 9uneq12i 3279 . 2  |-  ( dom 
A  u.  dom  B
)  =  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )
11 df-dm 4621 . 2  |-  dom  ( A  u.  B )  =  { y  |  E. x  y ( A  u.  B ) x }
127, 10, 113eqtr4ri 2202 1  |-  dom  ( A  u.  B )  =  ( dom  A  u.  dom  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 703    = wceq 1348   E.wex 1485   {cab 2156    u. cun 3119   class class class wbr 3989   dom cdm 4611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-br 3990  df-dm 4621
This theorem is referenced by:  rnun  5019  dmpropg  5083  dmtpop  5086  fntpg  5254  fnun  5304  sbthlemi5  6938  casedm  7063  djudm  7082  exmidfodomrlemim  7178  ennnfonelemhdmp1  12364  ennnfonelemkh  12367  strleund  12506  strleun  12507
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