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Definition df-iun 3717
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 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 3749. Theorem uniiun 3768 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 3715 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1286 . . . . 5 class 𝑦
76, 3wcel 1436 . . . 4 wff 𝑦𝐵
87, 1, 2wrex 2356 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2071 . 2 class {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
104, 9wceq 1287 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Colors of variables: wff set class
This definition is referenced by:  eliun  3719  nfiunxy  3741  nfiunya  3743  nfiu1  3745  dfiunv2  3751  cbviun  3752  iunss  3756  uniiun  3768  iunopab  4084  opeliunxp  4463  reliun  4528  fnasrn  5440  fnasrng  5442  abrexex2g  5850  abrexex2  5854  bdciun  11238
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