Theorem dmun 5819

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 4232	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)}
2		brun 5125	. . . . . 6 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
3	2	exbii 1850	. . . . 5 ⊢ (∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥))
4		19.43 1885	. . . . 5 ⊢ (∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥))
5	3, 4	bitr2i 275	. . . 4 ⊢ ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥)
6	5	abbii 2808	. . 3 ⊢ {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
7	1, 6	eqtri 2766	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
8		df-dm 5599	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
9		df-dm 5599	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
10	8, 9	uneq12i 4095	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
11		df-dm 5599	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥}
12	7, 10, 11	3eqtr4ri 2777	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 844   = wceq 1539  ∃wex 1782  {cab 2715   ∪ cun 3885   class class class wbr 5074  dom cdm 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-br 5075  df-dm 5599

This theorem is referenced by:  rnun  6049  dmpropg  6118  dmtpop  6121  fntpg  6494  fnun  6545  frrlem14  8115  wfrlem13OLD  8152  wfrlem16OLD  8155  tfrlem10  8218  sbthlem5  8874  fodomr  8915  axdc3lem4  10209  hashfun  14152  s4dom  14632  dmtrclfv  14729  strleun  16858  setsdm  16871  estrreslem2  17855  mvdco  19053  gsumzaddlem  19522  cnfldfunALT  20610  cnfldfunALTOLD  20611  uhgrun  27444  upgrun  27488  umgrun  27490  vtxdun  27848  wlkp1  28049  eupthp1  28580  bnj1416  33019  fineqvac  33066  satfdm  33331  fmlasuc0  33346  noextend  33869  noextendseq  33870  nosupbday  33908  nosupbnd1  33917  nosupbnd2  33919  noinfbday  33923  noinfbnd1  33932  noinfbnd2  33934  noetasuplem4  33939  noetainflem4  33943  fixun  34211  rclexi  41223  rtrclex  41225  rtrclexi  41229  cnvrcl0  41233  dmtrcl  41235  dfrtrcl5  41237  dfrcl2  41282  dmtrclfvRP  41338