ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmun GIF version

Theorem dmun 4793
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 3374 . . 3 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2 brun 4015 . . . . . 6 (𝑦(𝐴𝐵)𝑥 ↔ (𝑦𝐴𝑥𝑦𝐵𝑥))
32exbii 1585 . . . . 5 (∃𝑥 𝑦(𝐴𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥))
4 19.43 1608 . . . . 5 (∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
53, 4bitr2i 184 . . . 4 ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴𝐵)𝑥)
65abbii 2273 . . 3 {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
71, 6eqtri 2178 . 2 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
8 df-dm 4596 . . 3 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9 df-dm 4596 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
108, 9uneq12i 3259 . 2 (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11 df-dm 4596 . 2 dom (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
127, 10, 113eqtr4ri 2189 1 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wo 698   = wceq 1335  wex 1472  {cab 2143  cun 3100   class class class wbr 3965  dom cdm 4586
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-br 3966  df-dm 4596
This theorem is referenced by:  rnun  4994  dmpropg  5058  dmtpop  5061  fntpg  5226  fnun  5276  sbthlemi5  6905  casedm  7030  djudm  7049  exmidfodomrlemim  7136  ennnfonelemhdmp1  12149  ennnfonelemkh  12152  strleund  12289  strleun  12290
  Copyright terms: Public domain W3C validator