Definition df-iun 4714

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 4746. Theorem uniiun 4765 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6725 and funiunfv 6726 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 4712	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1636	. . . . 5 class 𝑦
7	6, 3	wcel 2156	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3097	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2792	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1637	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  4716  iuneq12df  4736  nfiun  4740  nfiu1  4742  dfiunv2  4748  cbviun  4749  iunss  4753  uniiun  4765  iunopab  5207  opeliunxp  5370  reliun  5441  fnasrn  6630  abrexex2g  7370  abrexex2OLD  7376  marypha2lem4  8579  cshwsiun  16014  cbviunf  29693  iuneq12daf  29694  iunrdx  29703  bnj956  31165  bnj1143  31179  bnj1146  31180  bnj1400  31224  bnj882  31314  bnj18eq1  31315  bnj893  31316  bnj1398  31420  volsupnfl  33762  ss2iundf  38445  iunssf  39750  opeliun2xp  42673  nfiund  42983