Definition df-iun 3966

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 disjoint variable group (meaning 𝐴 cannot depend on 𝑥) and that 𝐵 and 𝑥 do not share a disjoint 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 3998. Theorem uniiun 4018 provides a definition of ordinary union in terms of indexed union. (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 3964	. 2 class ∪ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1394	. . . . 5 class 𝑦
7	6, 3	wcel 2200	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wrex 2509	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2215	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1395	1 wff ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliun  3968  nfiunxy  3990  nfiunya  3992  nfiu1  3994  dfiunv2  4000  cbviun  4001  iunss  4005  uniiun  4018  iunopab  4369  opeliunxp  4773  reliun  4839  fnasrn  5812  fnasrng  5814  abrexex2g  6263  abrexex2  6267  bdciun  16199