Definition df-iun 4956

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 4993. Theorem uniiun 5020 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 7271 and funiunfv 7272 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 4954	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1569	. . . . 5 class 𝑦
7	6, 3	wcel 2147	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3090	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2741	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1570	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is used by:  eliun  4958  iuneq12df  4981  iuneq12d  4983  nfiun  4985  nfiung  4987  dfiun2g  4991  dfiunv2  4995  cbviun  4996  cbviung  4998  cbviunv  5000  iunssfOLD  5005  iunssOLD  5007  uniiun  5020  iunid  5022  iunsn  5027  iunopab  5538  opeliunxp  5722  opeliun2xp  5723  fnasrn  7169  abrexex2g  7997  marypha2lem4  9454  cshwsiun  17303  cbviunf  33219  iuneq12daf  33220  iunrdx  33227  iunrnmptss  33228  bnj956  35474  bnj1143  35487  bnj1146  35488  bnj1400  35532  bnj882  35623  bnj18eq1  35624  bnj893  35625  bnj1398  35731  iuneq12i  37161  cbviunvw2  37198  cbviundavw  37228  cbviundavw2  37252  ralssiun  38521  volsupnfl  38774  dfproplem  38832  iuneq1i  46304  nfiund  51060  nfiundg  51061