Definition df-iun 3715

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 3747. Theorem uniiun 3766 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 3713	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1286	. . . . 5 class y
7	6, 3	wcel 1436	. . . 4 wff y e. B
8	7, 1, 2	wrex 2356	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2071	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1287	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3717  nfiunxy  3739  nfiunya  3741  nfiu1  3743  dfiunv2  3749  cbviun  3750  iunss  3754  uniiun  3766  iunopab  4082  opeliunxp  4461  reliun  4526  fnasrn  5438  fnasrng  5440  abrexex2g  5848  abrexex2  5852  bdciun  11214