Definition df-iun 3918

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 3950. Theorem uniiun 3970 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 3916	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1363	. . . . 5 class y
7	6, 3	wcel 2167	. . . 4 wff y e. B
8	7, 1, 2	wrex 2476	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2182	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1364	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3920  nfiunxy  3942  nfiunya  3944  nfiu1  3946  dfiunv2  3952  cbviun  3953  iunss  3957  uniiun  3970  iunopab  4316  opeliunxp  4718  reliun  4784  fnasrn  5740  fnasrng  5742  abrexex2g  6177  abrexex2  6181  bdciun  15524