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Definition df-iun 3822
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, 𝑥𝐴 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 3854. Theorem uniiun 3873 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
df-iun 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Detailed syntax breakdown of Definition df-iun
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3ciun 3820 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1331 . . . . 5 class 𝑦
76, 3wcel 1481 . . . 4 wff 𝑦𝐵
87, 1, 2wrex 2418 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2126 . 2 class {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
104, 9wceq 1332 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Colors of variables: wff set class
This definition is referenced by:  eliun  3824  nfiunxy  3846  nfiunya  3848  nfiu1  3850  dfiunv2  3856  cbviun  3857  iunss  3861  uniiun  3873  iunopab  4210  opeliunxp  4601  reliun  4667  fnasrn  5605  fnasrng  5607  abrexex2g  6025  abrexex2  6029  bdciun  13245
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