Definition df-iun 4952

Description: Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, 𝐴 is independent of 𝑥 (although this is not required by the definition), and 𝐵 depends on 𝑥 i.e. can be read informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the index set, and 𝐵 the indexed set. In most books, 𝑥 ∈ 𝐴 is written as a subscript or underneath a union symbol ∪. We use a special union symbol ∪ to make it easier to distinguish from plain class union. In many theorems, you will see that 𝑥 and 𝐴 are in the same distinct variable group (meaning 𝐴 cannot depend on 𝑥) and that 𝐵 and 𝑥 do not share a distinct variable group (meaning that can be thought of as 𝐵(𝑥) i.e. can be substituted with a class expression containing 𝑥). An alternate definition tying indexed union to ordinary union is dfiun2 4989. Theorem uniiun 5016 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 7239 and funiunfv 7240 are useful when 𝐵 is a function. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Detailed syntax breakdown of Definition df-iun

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar 𝑥
2		cA	. . 3 class 𝐴
3		cB	. . 3 class 𝐵
4	1, 2, 3	ciun 4950	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1569	. . . . 5 class 𝑦
7	6, 3	wcel 2145	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3086	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2738	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1570	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is used by:  eliun  4954  iuneq12df  4977  iuneq12d  4979  nfiun  4981  nfiung  4983  dfiun2g  4987  dfiunv2  4991  cbviun  4992  cbviung  4994  cbviunv  4996  iunssfOLD  5001  iunssOLD  5003  uniiun  5016  iunid  5018  iunsn  5023  iunopab  5530  opeliunxp  5714  opeliun2xp  5715  fnasrn  7136  abrexex2g  7959  marypha2lem4  9408  cshwsiun  17238  cbviunf  33083  iuneq12daf  33084  iunrdx  33091  iunrnmptss  33092  bnj956  35341  bnj1143  35354  bnj1146  35355  bnj1400  35399  bnj882  35490  bnj18eq1  35491  bnj893  35492  bnj1398  35598  iuneq12i  36906  cbviunvw2  36943  cbviundavw  36973  cbviundavw2  36997  ralssiun  38250  volsupnfl  38503  dfproplem  38561  iuneq1i  46022  nfiund  50704  nfiundg  50705