Theorem dmun 4811

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 3389	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2		brun 4033	. . . . . 6 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
3	2	exbii 1593	. . . . 5 ⊢ (∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4		19.43 1616	. . . . 5 ⊢ (∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
5	3, 4	bitr2i 184	. . . 4 ⊢ ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥)
6	5	abbii 2282	. . 3 ⊢ {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
7	1, 6	eqtri 2186	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
8		df-dm 4614	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9		df-dm 4614	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
10	8, 9	uneq12i 3274	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11		df-dm 4614	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
12	7, 10, 11	3eqtr4ri 2197	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 698   = wceq 1343  ∃wex 1480  {cab 2151   ∪ cun 3114   class class class wbr 3982  dom cdm 4604

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-br 3983  df-dm 4614

This theorem is referenced by:  rnun  5012  dmpropg  5076  dmtpop  5079  fntpg  5244  fnun  5294  sbthlemi5  6926  casedm  7051  djudm  7070  exmidfodomrlemim  7157  ennnfonelemhdmp1  12342  ennnfonelemkh  12345  strleund  12483  strleun  12484