Definition df-iun 4457

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 4490. Theorem uniiun 4509 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6409 and funiunfv 6410 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 4455	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1474	. . . . 5 class 𝑦
7	6, 3	wcel 1977	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 2897	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2596	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1475	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  rabasiun  4459  eliun  4460  iuneq12df  4480  nfiun  4484  nfiu1  4486  dfiunv2  4492  cbviun  4493  iunss  4497  uniiun  4509  iunopab  4936  opeliunxp  5093  reliun  5162  fnasrn  6317  abrexex2g  7036  abrexex2  7040  marypha2lem4  8227  cshwsiun  15644  cbviunf  28755  iuneq12daf  28756  iunrdx  28764  bnj956  30101  bnj1143  30115  bnj1146  30116  bnj1400  30160  bnj882  30250  bnj18eq1  30251  bnj893  30252  bnj1398  30356  volsupnfl  32624  ss2iundf  36970  iunssf  38290  opeliun2xp  41904  nfiund  42219