Theorem dmun 4741

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 3338	. . 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 3974	. . . . . 6 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
3	2	exbii 1584	. . . . 5 |- ( E. x y ( A u. B ) x <-> E. x ( y A x \/ y B x ) )
4		19.43 1607	. . . . 5 |- ( E. x ( y A x \/ y B x ) <-> ( E. x y A x \/ E. x y B x ) )
5	3, 4	bitr2i 184	. . . 4 |- ( ( E. x y A x \/ E. x y B x ) <-> E. x y ( A u. B ) x )
6	5	abbii 2253	. . 3 |- { y | ( E. x y A x \/ E. x y B x ) } = { y | E. x y ( A u. B ) x }
7	1, 6	eqtri 2158	. 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 4544	. . 3 |- dom A = { y | E. x y A x }
9		df-dm 4544	. . 3 |- dom B = { y | E. x y B x }
10	8, 9	uneq12i 3223	. 2 |- ( dom A u. dom B ) = ( { y | E. x y A x } u. { y | E. x y B x } )
11		df-dm 4544	. 2 |- dom ( A u. B ) = { y | E. x y ( A u. B ) x }
12	7, 10, 11	3eqtr4ri 2169	1 |- dom ( A u. B ) = ( dom A u. dom B )

Syntax hints:   \/ wo 697   = wceq 1331   E.wex 1468   {cab 2123   u. cun 3064   class class class wbr 3924   dom cdm 4534

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-br 3925  df-dm 4544

This theorem is referenced by:  rnun  4942  dmpropg  5006  dmtpop  5009  fntpg  5174  fnun  5224  sbthlemi5  6842  casedm  6964  djudm  6983  exmidfodomrlemim  7050  ennnfonelemhdmp1  11911  ennnfonelemkh  11914  strleund  12036  strleun  12037