Theorem dmun 5902

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 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5113	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐴𝑥 ↔ 𝑧𝐴𝑥))
2	1	exbidv 1951	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑧𝐴𝑥))
3		breq1 5113	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑧𝐵𝑥))
4	3	exbidv 1951	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐵𝑥 ↔ ∃𝑥 𝑧𝐵𝑥))
5	2, 4	unabw 4261	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)}
6		brun 5163	. . . . . 6 ⊢ (𝑧(𝐴 ∪ 𝐵)𝑥 ↔ (𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
7	6	exbii 1878	. . . . 5 ⊢ (∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
8		19.43 1912	. . . . 5 ⊢ (∃𝑥(𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥) ↔ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥))
9	7, 8	bitr2i 279	. . . 4 ⊢ ((∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥) ↔ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥)
10	9	abbii 2830	. . 3 ⊢ {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)} = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
11	5, 10	eqtri 2786	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
12		df-dm 5673	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
13		df-dm 5673	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
14	12, 13	uneq12i 4121	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
15		df-dm 5673	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
16	11, 14, 15	3eqtr4ri 2797	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 860   = wceq 1570  ∃wex 1809  {cab 2741   ∪ cun 3904   class class class wbr 5110  dom cdm 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-dm 5673

This theorem is referenced by:  rnun  6144  dmpropg  6218  dmtpop  6221  fntpg  6598  fnun  6651  frrlem14  8297  tfrlem10  8375  sbthlem5  9080  fodomr  9117  fodomfir  9288  axdc3lem4  10438  hashfun  14476  s4dom  14958  dmtrclfv  15057  strleun  17218  setsdm  17231  estrreslem2  18195  mvdco  19516  gsumzaddlem  19992  cnfldfunALT  21518  noextend  27811  noextendseq  27812  nosupbday  27850  nosupbnd1  27859  nosupbnd2  27861  noinfbday  27865  noinfbnd1  27874  noinfbnd2  27876  noetasuplem4  27881  noetainflem4  27885  uhgrun  29405  upgrun  29449  umgrun  29451  vtxdun  29812  wlkp1  30010  eupthp1  30548  bnj1416  35408  fineqvac  35510  satfdm  35842  fmlasuc0  35857  fixun  36380  dmuncnvepres  39021  dfsucmap3  39093  rclexi  44324  rtrclex  44326  rtrclexi  44330  cnvrcl0  44334  dmtrcl  44336  dfrtrcl5  44338  dfrcl2  44383  dmtrclfvRP  44439