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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, x e. A is written as a subscript or underneath a union symbol U. . We use a special union symbol U_ 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 . Theorem uniiun provides a definition of ordinary union in terms of indexed union. Theorems fniunfv and funiunfv are useful when B is a function. (Contributed by NM, 27-Jun-1998)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-iun | |- U_ x e. A B = { y | E. x e. A y e. B } |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vx | |- x |
|
| 1 | cA | |- A |
|
| 2 | cB | |- B |
|
| 3 | 0 1 2 | ciun | |- U_ x e. A B |
| 4 | vy | |- y |
|
| 5 | 4 | cv | |- y |
| 6 | 5 2 | wcel | |- y e. B |
| 7 | 6 0 1 | wrex | |- E. x e. A y e. B |
| 8 | 7 4 | cab | |- { y | E. x e. A y e. B } |
| 9 | 3 8 | wceq | |- U_ x e. A B = { y | E. x e. A y e. B } |