Theorem dmun 4754

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 3348	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2		brun 3987	. . . . . 6 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
3	2	exbii 1585	. . . . 5 ⊢ (∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4		19.43 1608	. . . . 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 2256	. . 3 ⊢ {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
7	1, 6	eqtri 2161	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
8		df-dm 4557	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9		df-dm 4557	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
10	8, 9	uneq12i 3233	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11		df-dm 4557	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
12	7, 10, 11	3eqtr4ri 2172	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 698   = wceq 1332  ∃wex 1469  {cab 2126   ∪ cun 3074   class class class wbr 3937  dom cdm 4547

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-br 3938  df-dm 4557

This theorem is referenced by:  rnun  4955  dmpropg  5019  dmtpop  5022  fntpg  5187  fnun  5237  sbthlemi5  6857  casedm  6979  djudm  6998  exmidfodomrlemim  7074  ennnfonelemhdmp1  11958  ennnfonelemkh  11961  strleund  12086  strleun  12087