Theorem dmun 5905

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 5117	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐴𝑥 ↔ 𝑧𝐴𝑥))
2	1	exbidv 1954	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑧𝐴𝑥))
3		breq1 5117	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑧𝐵𝑥))
4	3	exbidv 1954	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐵𝑥 ↔ ∃𝑥 𝑧𝐵𝑥))
5	2, 4	unabw 4263	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)}
6		brun 5167	. . . . . 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 2833	. . 3 ⊢ {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)} = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
11	5, 10	eqtri 2789	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
12		df-dm 5676	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
13		df-dm 5676	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
14	12, 13	uneq12i 4123	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
15		df-dm 5676	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
16	11, 14, 15	3eqtr4ri 2800	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570  ∃wex 1812  {cab 2744   ∪ cun 3906   class class class wbr 5114  dom cdm 5666

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 2148  ax-9 2156  ax-ext 2738

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 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-dm 5676

This theorem is used by:  rnun  6147  dmpropg  6221  dmtpop  6224  fntpg  6603  fnun  6656  frrlem14  8305  tfrlem10  8383  sbthlem5  9089  fodomr  9126  fodomfir  9297  axdc3lem4  10455  hashfun  14494  s4dom  14982  dmtrclfv  15081  strleun  17242  setsdm  17255  estrreslem2  18219  mvdco  19546  gsumzaddlem  20022  cnfldfunALT  21574  noextend  27867  noextendseq  27868  nosupbday  27906  nosupbnd1  27915  nosupbnd2  27917  noinfbday  27921  noinfbnd1  27930  noinfbnd2  27932  noetasuplem4  27937  noetainflem4  27941  uhgrun  29461  upgrun  29505  umgrun  29507  vtxdun  29868  wlkp1  30066  eupthp1  30604  bnj1416  35459  fineqvac  35553  satfdm  35882  fmlasuc0  35897  fixun  36420  dmuncnvepres  39081  dfsucmap3  39153  rclexi  44382  rtrclex  44384  rtrclexi  44388  cnvrcl0  44392  dmtrcl  44394  dfrtrcl5  44396  dfrcl2  44441  dmtrclfvRP  44497