Theorem dmun 5857

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 5089	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐴𝑥 ↔ 𝑧𝐴𝑥))
2	1	exbidv 1923	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐴𝑥 ↔ ∃𝑥 𝑧𝐴𝑥))
3		breq1 5089	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦𝐵𝑥 ↔ 𝑧𝐵𝑥))
4	3	exbidv 1923	. . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 𝑦𝐵𝑥 ↔ ∃𝑥 𝑧𝐵𝑥))
5	2, 4	unabw 4248	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)}
6		brun 5137	. . . . . 6 ⊢ (𝑧(𝐴 ∪ 𝐵)𝑥 ↔ (𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
7	6	exbii 1850	. . . . 5 ⊢ (∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑧𝐴𝑥 ∨ 𝑧𝐵𝑥))
8		19.43 1884	. . . . 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 2804	. . 3 ⊢ {𝑧 ∣ (∃𝑥 𝑧𝐴𝑥 ∨ ∃𝑥 𝑧𝐵𝑥)} = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
11	5, 10	eqtri 2760	. 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
12		df-dm 5632	. . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥}
13		df-dm 5632	. . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}
14	12, 13	uneq12i 4107	. 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥})
15		df-dm 5632	. 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑧 ∣ ∃𝑥 𝑧(𝐴 ∪ 𝐵)𝑥}
16	11, 14, 15	3eqtr4ri 2771	1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)

Syntax hints:   ∨ wo 848   = wceq 1542  ∃wex 1781  {cab 2715   ∪ cun 3888   class class class wbr 5086  dom cdm 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-dm 5632

This theorem is referenced by:  rnun  6101  dmpropg  6171  dmtpop  6174  fntpg  6550  fnun  6604  frrlem14  8240  tfrlem10  8317  sbthlem5  9020  fodomr  9057  fodomfir  9229  axdc3lem4  10364  hashfun  14361  s4dom  14843  dmtrclfv  14942  strleun  17085  setsdm  17098  estrreslem2  18062  mvdco  19378  gsumzaddlem  19854  cnfldfunALT  21326  cnfldfunALTOLD  21339  noextend  27618  noextendseq  27619  nosupbday  27657  nosupbnd1  27666  nosupbnd2  27668  noinfbday  27672  noinfbnd1  27681  noinfbnd2  27683  noetasuplem4  27688  noetainflem4  27692  uhgrun  29131  upgrun  29175  umgrun  29177  vtxdun  29539  wlkp1  29737  eupthp1  30275  bnj1416  35187  fineqvac  35266  satfdm  35557  fmlasuc0  35572  fixun  36095  dmuncnvepres  38703  dfsucmap3  38775  rclexi  44045  rtrclex  44047  rtrclexi  44051  cnvrcl0  44055  dmtrcl  44057  dfrtrcl5  44059  dfrcl2  44104  dmtrclfvRP  44160