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Definition df-iun 4494
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 4527. Theorem uniiun 4546 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6470 and funiunfv 6471 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 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3ciun 4492 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1479 . . . . 5 class 𝑦
76, 3wcel 1987 . . . 4 wff 𝑦𝐵
87, 1, 2wrex 2909 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2607 . 2 class {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
104, 9wceq 1480 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Colors of variables: wff setvar class
This definition is referenced by:  rabasiun  4496  eliun  4497  iuneq12df  4517  nfiun  4521  nfiu1  4523  dfiunv2  4529  cbviun  4530  iunss  4534  uniiun  4546  iunopab  4982  opeliunxp  5141  reliun  5210  fnasrn  6376  abrexex2g  7105  abrexex2  7109  marypha2lem4  8304  cshwsiun  15749  cbviunf  29259  iuneq12daf  29260  iunrdx  29268  bnj956  30608  bnj1143  30622  bnj1146  30623  bnj1400  30667  bnj882  30757  bnj18eq1  30758  bnj893  30759  bnj1398  30863  volsupnfl  33125  ss2iundf  37471  iunssf  38785  opeliun2xp  41429  nfiund  41743
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