Theorem dmun 4988

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 4182	. . . . . 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 4784	. . 3 |- dom A = { y | E. x y A x }
9		df-dm 4784	. . 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 4784	. 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 4130   dom cdm 4774

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 4131  df-dm 4784

This theorem is used by:  rnun  5196  dmpropg  5260  dmtpop  5263  fntpg  5437  fnun  5489  sbthlemi5  7278  casedm  7426  djudm  7445  exmidfodomrlemim  7553  ennnfonelemhdmp1  13300  ennnfonelemkh  13303  bassetsnn  13409  strleund  13457  strleun  13458  uhgrun  16327  upgrun  16367  umgrun  16369  vtxdfifiun  16538