Definition df-iun 4554

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 4586. Theorem uniiun 4605 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6545 and funiunfv 6546 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 4552	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1522	. . . . 5 class 𝑦
7	6, 3	wcel 2030	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 2942	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2637	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1523	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  4556  iuneq12df  4576  nfiun  4580  nfiu1  4582  dfiunv2  4588  cbviun  4589  iunss  4593  uniiun  4605  iunopab  5041  opeliunxp  5204  reliun  5272  fnasrn  6451  abrexex2g  7186  abrexex2OLD  7192  marypha2lem4  8385  cshwsiun  15853  cbviunf  29498  iuneq12daf  29499  iunrdx  29508  bnj956  30973  bnj1143  30987  bnj1146  30988  bnj1400  31032  bnj882  31122  bnj18eq1  31123  bnj893  31124  bnj1398  31228  volsupnfl  33584  ss2iundf  38268  iunssf  39577  opeliun2xp  42436  nfiund  42746