Theorem dmun 5850

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 4258	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2		brun 5142	. . . . . 6 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
3	2	exbii 1849	. . . . 5 ⊢ (∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4		19.43 1883	. . . . 5 ⊢ (∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
5	3, 4	bitr2i 276	. . . 4 ⊢ ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥)
6	5	abbii 2798	. . 3 ⊢ {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
7	1, 6	eqtri 2754	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
8		df-dm 5626	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9		df-dm 5626	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
10	8, 9	uneq12i 4116	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11		df-dm 5626	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
12	7, 10, 11	3eqtr4ri 2765	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 847   = wceq 1541  ∃wex 1780  {cab 2709   ∪ cun 3900   class class class wbr 5091  dom cdm 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3907  df-br 5092  df-dm 5626

This theorem is referenced by:  rnun  6092  dmpropg  6162  dmtpop  6165  fntpg  6541  fnun  6595  frrlem14  8229  tfrlem10  8306  sbthlem5  9004  fodomr  9041  fodomfir  9212  axdc3lem4  10344  hashfun  14344  s4dom  14826  dmtrclfv  14925  strleun  17068  setsdm  17081  estrreslem2  18044  mvdco  19358  gsumzaddlem  19834  cnfldfunALT  21307  cnfldfunALTOLD  21320  noextend  27606  noextendseq  27607  nosupbday  27645  nosupbnd1  27654  nosupbnd2  27656  noinfbday  27660  noinfbnd1  27669  noinfbnd2  27671  noetasuplem4  27676  noetainflem4  27680  uhgrun  29053  upgrun  29097  umgrun  29099  vtxdun  29461  wlkp1  29659  eupthp1  30194  bnj1416  35049  fineqvac  35137  satfdm  35411  fmlasuc0  35426  fixun  35949  rclexi  43654  rtrclex  43656  rtrclexi  43660  cnvrcl0  43664  dmtrcl  43666  dfrtrcl5  43668  dfrcl2  43713  dmtrclfvRP  43769