Definition df-iun 4998

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 5035. Theorem uniiun 5060 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 7248 and funiunfv 7249 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 4996	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1538	. . . . 5 class 𝑦
7	6, 3	wcel 2104	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3068	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2707	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1539	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  5000  iuneq12df  5022  nfiun  5026  nfiung  5028  nfiu1  5030  dfiun2g  5032  dfiunv2  5037  cbviun  5038  cbviung  5040  iunssf  5046  iunss  5047  uniiun  5060  iunid  5062  iunsn  5068  iunopab  5558  iunopabOLD  5559  opeliunxp  5742  reliun  5815  fnasrn  7144  abrexex2g  7953  marypha2lem4  9435  cshwsiun  17037  cbviunf  32054  iuneq12daf  32055  iunrdx  32062  iunrnmptss  32064  bnj956  34085  bnj1143  34099  bnj1146  34100  bnj1400  34144  bnj882  34235  bnj18eq1  34236  bnj893  34237  bnj1398  34343  ralssiun  36591  volsupnfl  36836  opeliun2xp  47096  nfiund  47806  nfiundg  47807