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Definition df-iun 4124
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 4155. Theorem uniiun 4174 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6030 and funiunfv 6031 are useful when 𝐵 is a function. (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 set 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3ciun 4122 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 set 𝑦
65cv 1653 . . . . 5 class 𝑦
76, 3wcel 1728 . . . 4 wff 𝑦𝐵
87, 1, 2wrex 2713 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2429 . 2 class {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
104, 9wceq 1654 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Colors of variables: wff set class
This definition is referenced by:  eliun  4126  iuneq12df  4145  nfiun  4149  nfiu1  4151  dfiunv2  4157  cbviun  4158  iunss  4162  uniiun  4174  iunopab  4521  opeliunxp  4964  reliun  5030  fnasrn  5948  abrexex2g  6024  abrexex2  6037  marypha2lem4  7479  iuneq12daf  24043  iunrdx  24049  volsupnfl  26291  cshwsiun  28418  bnj956  29321  bnj1143  29335  bnj1146  29336  bnj1400  29381  bnj882  29471  bnj18eq1  29472  bnj893  29473  bnj1398  29577
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