Definition df-iun 3851

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 3883. Theorem uniiun 3902 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 3849	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1334	. . . . 5 class y
7	6, 3	wcel 2128	. . . 4 wff y e. B
8	7, 1, 2	wrex 2436	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2143	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1335	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3853  nfiunxy  3875  nfiunya  3877  nfiu1  3879  dfiunv2  3885  cbviun  3886  iunss  3890  uniiun  3902  iunopab  4241  opeliunxp  4640  reliun  4706  fnasrn  5644  fnasrng  5646  abrexex2g  6065  abrexex2  6069  bdciun  13440