Theorem dmun 4963

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 3488	. . 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 4161	. . . . . 6 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
3	2	exbii 1654	. . . . 5 |- ( E. x y ( A u. B ) x <-> E. x ( y A x \/ y B x ) )
4		19.43 1677	. . . . 5 |- ( E. x ( y A x \/ y B x ) <-> ( E. x y A x \/ E. x y B x ) )
5	3, 4	bitr2i 185	. . . 4 |- ( ( E. x y A x \/ E. x y B x ) <-> E. x y ( A u. B ) x )
6	5	abbii 2348	. . 3 |- { y | ( E. x y A x \/ E. x y B x ) } = { y | E. x y ( A u. B ) x }
7	1, 6	eqtri 2253	. 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 4759	. . 3 |- dom A = { y | E. x y A x }
9		df-dm 4759	. . 3 |- dom B = { y | E. x y B x }
10	8, 9	uneq12i 3371	. 2 |- ( dom A u. dom B ) = ( { y | E. x y A x } u. { y | E. x y B x } )
11		df-dm 4759	. 2 |- dom ( A u. B ) = { y | E. x y ( A u. B ) x }
12	7, 10, 11	3eqtr4ri 2264	1 |- dom ( A u. B ) = ( dom A u. dom B )

Syntax hints:   \/ wo 716   = wceq 1398   E.wex 1541   {cab 2218   u. cun 3209   class class class wbr 4109   dom cdm 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-br 4110  df-dm 4759

This theorem is referenced by:  rnun  5171  dmpropg  5235  dmtpop  5238  fntpg  5412  fnun  5464  sbthlemi5  7231  casedm  7377  djudm  7396  exmidfodomrlemim  7504  ennnfonelemhdmp1  13160  ennnfonelemkh  13163  bassetsnn  13269  strleund  13316  strleun  13317  uhgrun  16081  upgrun  16121  umgrun  16123  vtxdfifiun  16292