Definition df-iun 4451

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 4484. Theorem uniiun 4503 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6386 and funiunfv 6387 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 4449	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1473	. . . . 5 class 𝑦
7	6, 3	wcel 1976	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 2896	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2595	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1474	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  rabasiun  4453  eliun  4454  iuneq12df  4474  nfiun  4478  nfiu1  4480  dfiunv2  4486  cbviun  4487  iunss  4491  uniiun  4503  iunopab  4925  opeliunxp  5082  reliun  5150  fnasrn  6301  abrexex2g  7013  abrexex2  7017  marypha2lem4  8204  cshwsiun  15592  cbviunf  28548  iuneq12daf  28549  iunrdx  28557  bnj956  29894  bnj1143  29908  bnj1146  29909  bnj1400  29953  bnj882  30043  bnj18eq1  30044  bnj893  30045  bnj1398  30149  volsupnfl  32407  ss2iundf  36753  iunssf  38073  opeliun2xp  41885