Definition df-sup 4574

Description: Define the supremum of class B. It is meaningful when R is a relation that strictly orders A and when the supremum exists. For example, R could be 'less than', A could be the set of real numbers, and B could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrval 6671.

We will also use this notation for "infimum" by replacing R with `'R.

Assertion

Ref	Expression
df-sup	|- sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}

Distinct variable groups:   x,y,z,R   x,A,y,z   x,B,y,z

Detailed syntax breakdown of Definition df-sup

Step	Hyp	Ref	Expression
1		cB	. . 3 class B
2		cA	. . 3 class A
3		cR	. . 3 class R
4	1, 2, 3	csup 4573	. 2 class sup(B, A, R)
5		vx	. . . . . . . . 9 set x
6	5	cv 955	. . . . . . . 8 class x
7		vy	. . . . . . . . 9 set y
8	7	cv 955	. . . . . . . 8 class y
9	6, 8, 3	wbr 2619	. . . . . . 7 wff xRy
10	9	wn 2	. . . . . 6 wff -. xRy
11	10, 7, 1	wral 1645	. . . . 5 wff A.y e. B -. xRy
12	8, 6, 3	wbr 2619	. . . . . . 7 wff yRx
13		vz	. . . . . . . . . 10 set z
14	13	cv 955	. . . . . . . . 9 class z
15	8, 14, 3	wbr 2619	. . . . . . . 8 wff yRz
16	15, 13, 1	wrex 1646	. . . . . . 7 wff E.z e. B yRz
17	12, 16	wi 3	. . . . . 6 wff (yRx -> E.z e. B yRz)
18	17, 7, 2	wral 1645	. . . . 5 wff A.y e. A (yRx -> E.z e. B yRz)
19	11, 18	wa 223	. . . 4 wff (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))
20	19, 5, 2	crab 1648	. . 3 class {x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
21	20	cuni 2503	. 2 class U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
22	4, 21	wceq 956	1 wff sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}

This definition is referenced by:  supeq1 4575  supex 4577  supcl 4579  supub 4580  suplub 4581  suppr 4590  supsnALT 4592  lbinfm 6048  dfinfmr 6067  infmsup 6068  supxr 6081  supxrre 6083

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