Definition df-iun 4522

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 4554. Theorem uniiun 4573 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6505 and funiunfv 6506 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 4520	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1482	. . . . 5 class 𝑦
7	6, 3	wcel 1990	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 2913	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2608	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1483	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  4524  iuneq12df  4544  nfiun  4548  nfiu1  4550  dfiunv2  4556  cbviun  4557  iunss  4561  uniiun  4573  iunopab  5012  opeliunxp  5170  reliun  5239  fnasrn  6411  abrexex2g  7144  abrexex2OLD  7150  marypha2lem4  8344  cshwsiun  15806  cbviunf  29372  iuneq12daf  29373  iunrdx  29382  bnj956  30847  bnj1143  30861  bnj1146  30862  bnj1400  30906  bnj882  30996  bnj18eq1  30997  bnj893  30998  bnj1398  31102  volsupnfl  33454  ss2iundf  37951  iunssf  39263  opeliun2xp  42111  nfiund  42421