Definition df-oprab 6044

Description: Define the class abstraction (class builder) of a collection of nested ordered pairs (for use in defining operations). This is a special case of Definition 4.16 of [TakeutiZaring] p. 14. Normally x, y, and z are distinct, although the definition doesn't strictly require it. See df-ov 6043 for the value of an operation. The brace notation is called "class abstraction" by Quine; it is also called a "class builder" in the literature. The value of the most common operation class builder is given by ovmpt2 6168. (Contributed by NM, 12-Mar-1995.)

Assertion

Ref	Expression
df-oprab	|- { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }

Distinct variable groups:   x, w   y, w   z, w   ph, w

Allowed substitution hints:   ph( x, y, z)

Detailed syntax breakdown of Definition df-oprab

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		vx	. . 3 set x
3		vy	. . 3 set y
4		vz	. . 3 set z
5	1, 2, 3, 4	coprab 6041	. 2 class { <. <. x , y >. , z >. | ph }
6		vw	. . . . . . . . 9 set w
7	6	cv 1648	. . . . . . . 8 class w
8	2	cv 1648	. . . . . . . . . 10 class x
9	3	cv 1648	. . . . . . . . . 10 class y
10	8, 9	cop 3777	. . . . . . . . 9 class <. x , y >.
11	4	cv 1648	. . . . . . . . 9 class z
12	10, 11	cop 3777	. . . . . . . 8 class <. <. x , y >. , z >.
13	7, 12	wceq 1649	. . . . . . 7 wff w = <. <. x , y >. , z >.
14	13, 1	wa 359	. . . . . 6 wff ( w = <. <. x , y >. , z >. /\ ph )
15	14, 4	wex 1547	. . . . 5 wff E. z ( w = <. <. x , y >. , z >. /\ ph )
16	15, 3	wex 1547	. . . 4 wff E. y E. z ( w = <. <. x , y >. , z >. /\ ph )
17	16, 2	wex 1547	. . 3 wff E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph )
18	17, 6	cab 2390	. 2 class { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }
19	5, 18	wceq 1649	1 wff { <. <. x , y >. , z >. | ph } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ph ) }

This definition is referenced by:  oprabid  6064  dfoprab2  6080  nfoprab1  6082  nfoprab2  6083  nfoprab3  6084  nfoprab  6085  oprabbid  6086  ssoprab2  6089  cbvoprab2  6104  eloprabga  6119  oprabrexex2  6148  eloprabi  6372  mpt20  6386  dftpos3  6456  colinearex  25898