Theorem dmun 4990

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 3498	. . 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 4183	. . . . . 6 |- ( y ( A u. B ) x <-> ( y A x \/ y B x ) )
3	2	exbii 1658	. . . . 5 |- ( E. x y ( A u. B ) x <-> E. x ( y A x \/ y B x ) )
4		19.43 1681	. . . . 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 2354	. . 3 |- { y | ( E. x y A x \/ E. x y B x ) } = { y | E. x y ( A u. B ) x }
7	1, 6	eqtri 2259	. 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 4786	. . 3 |- dom A = { y | E. x y A x }
9		df-dm 4786	. . 3 |- dom B = { y | E. x y B x }
10	8, 9	uneq12i 3381	. 2 |- ( dom A u. dom B ) = ( { y | E. x y A x } u. { y | E. x y B x } )
11		df-dm 4786	. 2 |- dom ( A u. B ) = { y | E. x y ( A u. B ) x }
12	7, 10, 11	3eqtr4ri 2270	1 |- dom ( A u. B ) = ( dom A u. dom B )

This proof depends on syntax axioms:   \/ wo 720   = wceq 1402   E.wex 1545   {cab 2224   u. cun 3218   class class class wbr 4131   dom cdm 4776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-br 4132  df-dm 4786

This theorem is used by:  rnun  5199  dmpropg  5263  dmtpop  5266  fntpg  5440  fnun  5492  sbthlemi5  7283  casedm  7432  djudm  7451  exmidfodomrlemim  7559  ennnfonelemhdmp1  13358  ennnfonelemkh  13361  bassetsnn  13467  strleund  13516  strleun  13517  uhgrun  16532  upgrun  16572  umgrun  16574  vtxdfifiun  16743