Definition df-iun 3890

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 3922. Theorem uniiun 3942 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 3888	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1352	. . . . 5 class y
7	6, 3	wcel 2148	. . . 4 wff y e. B
8	7, 1, 2	wrex 2456	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2163	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1353	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3892  nfiunxy  3914  nfiunya  3916  nfiu1  3918  dfiunv2  3924  cbviun  3925  iunss  3929  uniiun  3942  iunopab  4283  opeliunxp  4683  reliun  4749  fnasrn  5696  fnasrng  5698  abrexex2g  6123  abrexex2  6127  bdciun  14715