Definition df-iun 3706

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 3738. Theorem uniiun 3757 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 3704	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1284	. . . . 5 class y
7	6, 3	wcel 1434	. . . 4 wff y e. B
8	7, 1, 2	wrex 2354	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2069	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1285	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3708  nfiunxy  3730  nfiunya  3732  nfiu1  3734  dfiunv2  3740  cbviun  3741  iunss  3745  uniiun  3757  iunopab  4072  opeliunxp  4451  reliun  4516  fnasrn  5417  fnasrng  5419  abrexex2g  5826  abrexex2  5830  bdciun  11112