Definition df-iun 4946

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 4985. Theorem uniiun 5012 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 7191 and funiunfv 7192 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 4944	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1540	. . . . 5 class 𝑦
7	6, 3	wcel 2113	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3058	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2712	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1541	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  4948  iuneq12df  4971  iuneq12d  4974  nfiun  4976  nfiung  4978  nfiu1OLD  4981  dfiun2g  4983  dfiunv2  4987  cbviun  4988  cbviung  4990  cbviunv  4992  iunssfOLD  4997  iunssOLD  4999  uniiun  5012  iunid  5014  iunsn  5019  iunopab  5505  opeliunxp  5689  opeliun2xp  5690  fnasrn  7088  abrexex2g  7906  marypha2lem4  9339  cshwsiun  17025  cbviunf  32579  iuneq12daf  32580  iunrdx  32587  iunrnmptss  32589  bnj956  34881  bnj1143  34895  bnj1146  34896  bnj1400  34940  bnj882  35031  bnj18eq1  35032  bnj893  35033  bnj1398  35139  iuneq12i  36338  cbviunvw2  36375  cbviundavw  36405  cbviundavw2  36429  ralssiun  37551  volsupnfl  37805  iuneq1i  45271  nfiund  49861  nfiundg  49862