Definition df-iun 4666

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 4698. Theorem uniiun 4717 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6660 and funiunfv 6661 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 4664	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1623	. . . . 5 class 𝑦
7	6, 3	wcel 2131	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3043	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2738	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1624	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  4668  iuneq12df  4688  nfiun  4692  nfiu1  4694  dfiunv2  4700  cbviun  4701  iunss  4705  uniiun  4717  iunopab  5154  opeliunxp  5319  reliun  5387  fnasrn  6566  abrexex2g  7301  abrexex2OLD  7307  marypha2lem4  8501  cshwsiun  16000  cbviunf  29671  iuneq12daf  29672  iunrdx  29681  bnj956  31146  bnj1143  31160  bnj1146  31161  bnj1400  31205  bnj882  31295  bnj18eq1  31296  bnj893  31297  bnj1398  31401  volsupnfl  33759  ss2iundf  38445  iunssf  39754  opeliun2xp  42613  nfiund  42923