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Definition df-iun 4451
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 4484. Theorem uniiun 4503 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6386 and funiunfv 6387 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 4449 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1473 . . . . 5 class 𝑦
76, 3wcel 1976 . . . 4 wff 𝑦𝐵
87, 1, 2wrex 2896 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2595 . 2 class {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
104, 9wceq 1474 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Colors of variables: wff setvar class
This definition is referenced by:  rabasiun  4453  eliun  4454  iuneq12df  4474  nfiun  4478  nfiu1  4480  dfiunv2  4486  cbviun  4487  iunss  4491  uniiun  4503  iunopab  4925  opeliunxp  5082  reliun  5150  fnasrn  6301  abrexex2g  7013  abrexex2  7017  marypha2lem4  8204  cshwsiun  15592  cbviunf  28548  iuneq12daf  28549  iunrdx  28557  bnj956  29894  bnj1143  29908  bnj1146  29909  bnj1400  29953  bnj882  30043  bnj18eq1  30044  bnj893  30045  bnj1398  30149  volsupnfl  32407  ss2iundf  36753  iunssf  38073  opeliun2xp  41885
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