Theorem dmun 4560

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.

Step	Hyp	Ref	Expression
1		unab 3231	. . 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 3831	. . . . . 6 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
3	2	exbii 1536	. . . . 5 |- ( E. x y ( A u. B ) x <-> E. x ( y A x \/ y B x ) )
4		19.43 1559	. . . . 5 |- ( E. x ( y A x \/ y B x ) <-> ( E. x y A x \/ E. x y B x ) )
5	3, 4	bitr2i 183	. . . 4 |- ( ( E. x y A x \/ E. x y B x ) <-> E. x y ( A u. B ) x )
6	5	abbii 2194	. . 3 |- { y | ( E. x y A x \/ E. x y B x ) } = { y | E. x y ( A u. B ) x }
7	1, 6	eqtri 2101	. 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 4373	. . 3 |- dom A = { y | E. x y A x }
9		df-dm 4373	. . 3 |- dom B = { y | E. x y B x }
10	8, 9	uneq12i 3124	. 2 |- ( dom A u. dom B ) = ( { y | E. x y A x } u. { y | E. x y B x } )
11		df-dm 4373	. 2 |- dom ( A u. B ) = { y | E. x y ( A u. B ) x }
12	7, 10, 11	3eqtr4ri 2112	1 |- dom ( A u. B ) = ( dom A u. dom B )

Syntax hints:   \/ wo 661   = wceq 1284   E.wex 1421   {cab 2067   u. cun 2971   class class class wbr 3785   dom cdm 4363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-br 3786  df-dm 4373

This theorem is referenced by:  rnun  4752  dmpropg  4813  dmtpop  4816  fntpg  4975  fnun  5025