Definition df-iun 3257

Description: Define indexed union. Definition of [Stoll] p. 45. In normal use, A is independent of x, 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 3285. Theorem uniiun 3306 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 4841 and funiunfv 4842 are useful when B is a function.

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

Detailed syntax breakdown of Definition df-iun

Step	Hyp	Ref	Expression
1		vx	. . 3 set x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	ciun 3255	. 2 class U_x e. A B
5		vy	. . . . . 6 set y
6	5	cv 1297	. . . . 5 class y
7	6, 3	wcel 1300	. . . 4 wff y e. B
8	7, 1, 2	wrex 2106	. . 3 wff E.x e. A y e. B
9	8, 5	cab 1871	. 2 class {y | E.x e. A y e. B}
10	4, 9	wceq 1298	1 wff U_x e. A B = {y | E.x e. A y e. B}

This definition is referenced by:  eliun 3259  iunss1OLD 3267  hbiu1 3281  dfiun2gOLD 3284  iunss 3291  uniiun 3306  iunun 3328  reliun 4101  iunfopab 4542  abrexex2 4847  bnj884 12809  bnj898 12815  bnj956 12853  bnj1144 12941  bnj1146 12943  bnj1401 13113  bnj1398 13515  bnj1499 13561  abrexex2g 15738

Copyright terms: Public domain