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Theorem dmun 4818
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 (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)

Proof of Theorem dmun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unab 3394 . . 3 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2 brun 4040 . . . . . 6 (𝑦(𝐴𝐵)𝑥 ↔ (𝑦𝐴𝑥𝑦𝐵𝑥))
32exbii 1598 . . . . 5 (∃𝑥 𝑦(𝐴𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥))
4 19.43 1621 . . . . 5 (∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
53, 4bitr2i 184 . . . 4 ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴𝐵)𝑥)
65abbii 2286 . . 3 {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
71, 6eqtri 2191 . 2 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
8 df-dm 4621 . . 3 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9 df-dm 4621 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
108, 9uneq12i 3279 . 2 (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11 df-dm 4621 . 2 dom (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
127, 10, 113eqtr4ri 2202 1 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wo 703   = wceq 1348  wex 1485  {cab 2156  cun 3119   class class class wbr 3989  dom cdm 4611
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-br 3990  df-dm 4621
This theorem is referenced by:  rnun  5019  dmpropg  5083  dmtpop  5086  fntpg  5254  fnun  5304  sbthlemi5  6938  casedm  7063  djudm  7082  exmidfodomrlemim  7178  ennnfonelemhdmp1  12364  ennnfonelemkh  12367  strleund  12506  strleun  12507
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