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	|- U_ x e. A B = { y | E. x e. A y e. B }

Detailed syntax breakdown

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 }