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Theorem dmun 4792
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 3342 . . 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 3966 . . . . . 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 2361 . . 3  |-  { y  |  ( E. x  y A x  \/  E. x  y B x ) }  =  {
y  |  E. x  y ( A  u.  B ) x }
71, 6eqtri 2273 . 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 4598 . . 3  |-  dom  A  =  { y  |  E. x  y A x }
9 df-dm 4598 . . 3  |-  dom  B  =  { y  |  E. x  y B x }
108, 9uneq12i 3237 . 2  |-  ( dom 
A  u.  dom  B
)  =  ( { y  |  E. x  y A x }  u.  { y  |  E. x  y B x } )
11 df-dm 4598 . 2  |-  dom  (  A  u.  B )  =  { y  |  E. x  y ( A  u.  B ) x }
127, 10, 113eqtr4ri 2284 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 2239    u. cun 3076   class class class wbr 3920   dom cdm 4580
This theorem is referenced by:  rnun  4996  dmpropg  5052  dmtpop  5055  fnun  5207  tfrlem10  6289  sbthlem5  6860  fodomr  6897  axdc3lem4  7963  hashfun  11266  strlemor1  13109  strleun  13112  xpsfrnel2  13341  wfrlem13  23436  wfrlem16  23439  fixun  23624  mvdco  26554  bnj1416  27758
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-br 3921  df-dm 4598
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