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Definition df-iun 3903
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 3935. Theorem uniiun 3955 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 3901 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1363 . . . . 5 class 𝑦
76, 3wcel 2160 . . . 4 wff 𝑦𝐵
87, 1, 2wrex 2469 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2175 . 2 class {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
104, 9wceq 1364 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Colors of variables: wff set class
This definition is referenced by:  eliun  3905  nfiunxy  3927  nfiunya  3929  nfiu1  3931  dfiunv2  3937  cbviun  3938  iunss  3942  uniiun  3955  iunopab  4299  opeliunxp  4699  reliun  4765  fnasrn  5715  fnasrng  5717  abrexex2g  6146  abrexex2  6150  bdciun  15108
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