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Theorem dmun 4834
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 3402 . . 3 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2 brun 4054 . . . . . 6 (𝑦(𝐴𝐵)𝑥 ↔ (𝑦𝐴𝑥𝑦𝐵𝑥))
32exbii 1605 . . . . 5 (∃𝑥 𝑦(𝐴𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥))
4 19.43 1628 . . . . 5 (∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
53, 4bitr2i 185 . . . 4 ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴𝐵)𝑥)
65abbii 2293 . . 3 {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
71, 6eqtri 2198 . 2 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
8 df-dm 4636 . . 3 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9 df-dm 4636 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
108, 9uneq12i 3287 . 2 (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11 df-dm 4636 . 2 dom (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
127, 10, 113eqtr4ri 2209 1 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wo 708   = wceq 1353  wex 1492  {cab 2163  cun 3127   class class class wbr 4003  dom cdm 4626
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 2739  df-un 3133  df-br 4004  df-dm 4636
This theorem is referenced by:  rnun  5037  dmpropg  5101  dmtpop  5104  fntpg  5272  fnun  5322  sbthlemi5  6959  casedm  7084  djudm  7103  exmidfodomrlemim  7199  ennnfonelemhdmp1  12409  ennnfonelemkh  12412  strleund  12561  strleun  12562
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