Metamath Proof Explorer

Definition df-iun

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 x A B = y | x A y B

Detailed syntax breakdown

Step Hyp Ref Expression
0 vx setvar x
1 cA class A
2 cB class B
3 0 1 2 ciun class x A B
4 vy setvar y
5 4 cv setvar y
6 5 2 wcel wff y B
7 6 0 1 wrex wff x A y B
8 7 4 cab class y | x A y B
9 3 8 wceq wff x A B = y | x A y B