Theorem dmun 4835

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 ∪ 𝐵) = (dom 𝐴 ∪ 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 3403	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2		brun 4055	. . . . . 6 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
3	2	exbii 1605	. . . . 5 ⊢ (∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4		19.43 1628	. . . . 5 ⊢ (∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
5	3, 4	bitr2i 185	. . . 4 ⊢ ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥)
6	5	abbii 2293	. . 3 ⊢ {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
7	1, 6	eqtri 2198	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
8		df-dm 4637	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9		df-dm 4637	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
10	8, 9	uneq12i 3288	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11		df-dm 4637	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
12	7, 10, 11	3eqtr4ri 2209	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 708   = wceq 1353  ∃wex 1492  {cab 2163   ∪ cun 3128   class class class wbr 4004  dom cdm 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-br 4005  df-dm 4637

This theorem is referenced by:  rnun  5038  dmpropg  5102  dmtpop  5105  fntpg  5273  fnun  5323  sbthlemi5  6960  casedm  7085  djudm  7104  exmidfodomrlemim  7200  ennnfonelemhdmp1  12410  ennnfonelemkh  12413  strleund  12562  strleun  12563