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Theorem dmun 5474
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 4025 . . 3 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2 brun 4843 . . . . . 6 (𝑦(𝐴𝐵)𝑥 ↔ (𝑦𝐴𝑥𝑦𝐵𝑥))
32exbii 1911 . . . . 5 (∃𝑥 𝑦(𝐴𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥))
4 19.43 1947 . . . . 5 (∃𝑥(𝑦𝐴𝑥𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
53, 4bitr2i 265 . . . 4 ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴𝐵)𝑥)
65abbii 2865 . . 3 {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
71, 6eqtri 2770 . 2 ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
8 df-dm 5264 . . 3 dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9 df-dm 5264 . . 3 dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
108, 9uneq12i 3896 . 2 (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11 df-dm 5264 . 2 dom (𝐴𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴𝐵)𝑥}
127, 10, 113eqtr4ri 2781 1 dom (𝐴𝐵) = (dom 𝐴 ∪ dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wo 382   = wceq 1620  wex 1841  {cab 2734  cun 3701   class class class wbr 4792  dom cdm 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-un 3708  df-br 4793  df-dm 5264
This theorem is referenced by:  rnun  5687  dmpropg  5755  dmtpop  5758  fntpg  6097  fnun  6146  wfrlem13  7584  wfrlem16  7587  tfrlem10  7640  sbthlem5  8227  fodomr  8264  axdc3lem4  9438  hashfun  13387  s4dom  13835  dmtrclfv  13929  setsdm  16065  strlemor1OLD  16142  strleun  16145  xpsfrnel2  16398  estrreslem2  16950  mvdco  18036  gsumzaddlem  18492  cnfldfun  19931  uhgrun  26139  upgrun  26183  umgrun  26185  vtxdun  26558  wlkp1  26759  eupthp1  27339  bnj1416  31385  noextend  32096  noextendseq  32097  nosupbday  32128  nosupbnd1  32137  nosupbnd2  32139  noetalem3  32142  noetalem4  32143  fixun  32293  rclexi  38393  rtrclex  38395  rtrclexi  38399  cnvrcl0  38403  dmtrcl  38405  dfrtrcl5  38407  dfrcl2  38437  dmtrclfvRP  38493
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