Definition df-iun 4014

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 4046. Theorem uniiun 4066 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 4012	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1401	. . . . 5 class y
7	6, 3	wcel 2209	. . . 4 wff y e. B
8	7, 1, 2	wrex 2529	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2224	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1402	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is used by:  eliun  4016  nfiunxy  4038  nfiunya  4040  nfiu1  4042  dfiunv2  4048  cbviun  4049  iunss  4053  uniiun  4066  iunopab  4424  opeliunxp  4830  reliun  4898  fnasrn  5887  fnasrng  5889  abrexex2g  6349  abrexex2  6353  bdciun  16904