Theorem dmun 4913

Description: The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 12-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
dmun	⊢ dom (A ∪ B) = (dom A ∪ dom B)

Proof of Theorem dmun

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdm3 4902	. 2 ⊢ dom (A ∪ B) = {x ∣ ∃y⟨x, y⟩ ∈ (A ∪ B)}
2		eldm 4899	. . . . 5 ⊢ (x ∈ dom A ↔ ∃y xAy)
3		eldm 4899	. . . . 5 ⊢ (x ∈ dom B ↔ ∃y xBy)
4	2, 3	orbi12i 507	. . . 4 ⊢ ((x ∈ dom A ∨ x ∈ dom B) ↔ (∃y xAy ∨ ∃y xBy))
5		elun 3221	. . . 4 ⊢ (x ∈ (dom A ∪ dom B) ↔ (x ∈ dom A ∨ x ∈ dom B))
6		df-br 4641	. . . . . . 7 ⊢ (x(A ∪ B)y ↔ ⟨x, y⟩ ∈ (A ∪ B))
7		brun 4693	. . . . . . 7 ⊢ (x(A ∪ B)y ↔ (xAy ∨ xBy))
8	6, 7	bitr3i 242	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (A ∪ B) ↔ (xAy ∨ xBy))
9	8	exbii 1582	. . . . 5 ⊢ (∃y⟨x, y⟩ ∈ (A ∪ B) ↔ ∃y(xAy ∨ xBy))
10		19.43 1605	. . . . 5 ⊢ (∃y(xAy ∨ xBy) ↔ (∃y xAy ∨ ∃y xBy))
11	9, 10	bitri 240	. . . 4 ⊢ (∃y⟨x, y⟩ ∈ (A ∪ B) ↔ (∃y xAy ∨ ∃y xBy))
12	4, 5, 11	3bitr4i 268	. . 3 ⊢ (x ∈ (dom A ∪ dom B) ↔ ∃y⟨x, y⟩ ∈ (A ∪ B))
13	12	abbi2i 2465	. 2 ⊢ (dom A ∪ dom B) = {x ∣ ∃y⟨x, y⟩ ∈ (A ∪ B)}
14	1, 13	eqtr4i 2376	1 ⊢ dom (A ∪ B) = (dom A ∪ dom B)

Syntax hints:   ∨ wo 357  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ∪ cun 3208  ⟨cop 4562   class class class wbr 4640  dom cdm 4773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by:  rnun  5037  dmpropg  5069  dmtpop  5072  fnun  5190