Definition df-iun 4821

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 4855. Theorem uniiun 4875 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6862 and funiunfv 6863 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 4819	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1519	. . . . 5 class 𝑦
7	6, 3	wcel 2079	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 3104	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2773	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1520	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  4823  iuneq12df  4844  nfiun  4848  nfiu1  4850  dfiunv2  4857  cbviun  4858  iunss  4862  uniiun  4875  iunopab  5326  opeliunxp  5497  reliun  5567  fnasrn  6761  abrexex2g  7512  marypha2lem4  8738  cshwsiun  16250  cbviunf  29976  iuneq12daf  29977  iunrdx  29984  iunrnmptss  29986  bnj956  31621  bnj1143  31635  bnj1146  31636  bnj1400  31680  bnj882  31770  bnj18eq1  31771  bnj893  31772  bnj1398  31876  ralssiun  34165  volsupnfl  34414  iunsn  38598  ss2iundf  39440  iunssf  40843  opeliun2xp  43813  nfiund  44212