Definition df-iun 3995

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 disjoint variable group (meaning A cannot depend on x) and that B and x do not share a disjoint 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 4027. Theorem uniiun 4047 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
df-iun	|- U_ x e. A B = { y | E. x e. A y e. B }

Distinct variable groups:   x, y   y, A   y, B

Allowed substitution hints:   A( x)   B( x)

Detailed syntax breakdown of Definition df-iun

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	ciun 3993	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1397	. . . . 5 class y
7	6, 3	wcel 2205	. . . 4 wff y e. B
8	7, 1, 2	wrex 2523	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2220	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1398	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3997  nfiunxy  4019  nfiunya  4021  nfiu1  4023  dfiunv2  4029  cbviun  4030  iunss  4034  uniiun  4047  iunopab  4402  opeliunxp  4807  reliun  4875  fnasrn  5858  fnasrng  5860  abrexex2g  6315  abrexex2  6319  bdciun  16665