Definition df-iun 4953

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 4991. Theorem uniiun 5018 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 7233 and funiunfv 7234 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 4951	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1561	. . . . 5 class 𝑦
7	6, 3	wcel 2144	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3088	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2742	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1562	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  4955  iuneq12df  4978  iuneq12d  4981  nfiun  4983  nfiung  4985  dfiun2g  4989  dfiunv2  4993  cbviun  4994  cbviung  4996  cbviunv  4998  iunssfOLD  5003  iunssOLD  5005  uniiun  5018  iunid  5020  iunsn  5025  iunopab  5532  opeliunxp  5716  opeliun2xp  5717  fnasrn  7129  abrexex2g  7947  marypha2lem4  9386  cshwsiun  17137  cbviunf  32757  iuneq12daf  32758  iunrdx  32765  iunrnmptss  32767  bnj956  35074  bnj1143  35087  bnj1146  35088  bnj1400  35132  bnj882  35223  bnj18eq1  35224  bnj893  35225  bnj1398  35331  iuneq12i  36560  cbviunvw2  36597  cbviundavw  36627  cbviundavw2  36651  ralssiun  37906  volsupnfl  38169  iuneq1i  45669  nfiund  50300  nfiundg  50301