Theorem dmun 5856

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 5098	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐴𝑥 ↔ 𝑧𝐴𝑥))
2	1	exbidv 1922	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑧𝐴𝑥))
3		breq1 5098	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑧𝐵𝑥))
4	3	exbidv 1922	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐵𝑥 ↔ ∃𝑥 𝑧𝐵𝑥))
5	2, 4	unabw 4256	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)}
6		brun 5146	. . . . . 6 ⊢ (𝑧(𝐴 ∪ 𝐵)𝑥 ↔ (𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
7	6	exbii 1849	. . . . 5 ⊢ (∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
8		19.43 1883	. . . . 5 ⊢ (∃𝑥(𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥) ↔ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥))
9	7, 8	bitr2i 276	. . . 4 ⊢ ((∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥) ↔ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥)
10	9	abbii 2800	. . 3 ⊢ {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)} = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
11	5, 10	eqtri 2756	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
12		df-dm 5631	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
13		df-dm 5631	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
14	12, 13	uneq12i 4115	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
15		df-dm 5631	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
16	11, 14, 15	3eqtr4ri 2767	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 847   = wceq 1541  ∃wex 1780  {cab 2711   ∪ cun 3896   class class class wbr 5095  dom cdm 5621

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 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-dm 5631

This theorem is referenced by:  rnun  6099  dmpropg  6169  dmtpop  6172  fntpg  6548  fnun  6602  frrlem14  8237  tfrlem10  8314  sbthlem5  9013  fodomr  9050  fodomfir  9221  axdc3lem4  10353  hashfun  14348  s4dom  14830  dmtrclfv  14929  strleun  17072  setsdm  17085  estrreslem2  18048  mvdco  19361  gsumzaddlem  19837  cnfldfunALT  21310  cnfldfunALTOLD  21323  noextend  27608  noextendseq  27609  nosupbday  27647  nosupbnd1  27656  nosupbnd2  27658  noinfbday  27662  noinfbnd1  27671  noinfbnd2  27673  noetasuplem4  27678  noetainflem4  27682  uhgrun  29056  upgrun  29100  umgrun  29102  vtxdun  29464  wlkp1  29662  eupthp1  30200  bnj1416  35074  fineqvac  35162  satfdm  35436  fmlasuc0  35451  fixun  35974  dmuncnvepres  38438  dfsucmap3  38499  rclexi  43735  rtrclex  43737  rtrclexi  43741  cnvrcl0  43745  dmtrcl  43747  dfrtrcl5  43749  dfrcl2  43794  dmtrclfvRP  43850