Definition df-iun 3867

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 3899. Theorem uniiun 3918 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 3865	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1342	. . . . 5 class y
7	6, 3	wcel 2136	. . . 4 wff y e. B
8	7, 1, 2	wrex 2444	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2151	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1343	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3869  nfiunxy  3891  nfiunya  3893  nfiu1  3895  dfiunv2  3901  cbviun  3902  iunss  3906  uniiun  3918  iunopab  4258  opeliunxp  4658  reliun  4724  fnasrn  5662  fnasrng  5664  abrexex2g  6085  abrexex2  6089  bdciun  13720