Theorem dmun 5892

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 5106	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐴𝑥 ↔ 𝑧𝐴𝑥))
2	1	exbidv 1954	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑧𝐴𝑥))
3		breq1 5106	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑧𝐵𝑥))
4	3	exbidv 1954	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐵𝑥 ↔ ∃𝑥 𝑧𝐵𝑥))
5	2, 4	unabw 4253	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)}
6		brun 5156	. . . . . 6 ⊢ (𝑧(𝐴 ∪ 𝐵)𝑥 ↔ (𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
7	6	exbii 1881	. . . . 5 ⊢ (∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
8		19.43 1915	. . . . 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 2828	. . 3 ⊢ {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)} = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
11	5, 10	eqtri 2784	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
12		df-dm 5661	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
13		df-dm 5661	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
14	12, 13	uneq12i 4113	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
15		df-dm 5661	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
16	11, 14, 15	3eqtr4ri 2795	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570  ∃wex 1812  {cab 2739   ∪ cun 3897   class class class wbr 5103  dom cdm 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-dm 5661

This theorem is used by:  rnun  6134  dmpropg  6209  dmtpop  6212  fntpg  6592  fnun  6645  frrlem14  8301  tfrlem10  8379  sbthlem5  9094  fodomr  9131  fodomfir  9303  axdc3lem4  10512  hashfun  14562  s4dom  15050  dmtrclfv  15151  strleun  17315  setsdm  17328  estrreslem2  18292  mvdco  19639  gsumzaddlem  20115  cnfldfunALT  21673  noextend  28005  noextendseq  28006  nosupbday  28044  nosupbnd1  28053  nosupbnd2  28055  noinfbday  28059  noinfbnd1  28068  noinfbnd2  28070  noetasuplem4  28075  noetainflem4  28079  uhgrun  29634  upgrun  29678  umgrun  29680  vtxdun  30044  wlkp1  30242  eupthp1  30799  bnj1416  35652  fineqvac  35757  satfdm  36103  fmlasuc0  36118  fixun  36641  dmuncnvepres  39291  dfsucmap3  39363  rclexi  44574  rtrclex  44576  rtrclexi  44580  cnvrcl0  44584  dmtrcl  44586  dfrtrcl5  44588  dfrcl2  44633  dmtrclfvRP  44689