Definition df-right 31934

Description: Define the left options of a surreal. This is the set of surreals that are "closest" on the right to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)

Assertion

Ref	Expression
df-right	|- R = ( x e. No |-> { y e. ( O ` ( bday ` x ) ) | A. z e. No ( ( x <s z /\ z <s y ) -> ( bday ` y ) e. ( bday ` z ) ) } )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-right

Step	Hyp	Ref	Expression
1		cright 31929	. 2 class R
2		vx	. . 3 setvar x
3		csur 31793	. . 3 class No
4	2	cv 1482	. . . . . . . 8 class x
5		vz	. . . . . . . . 9 setvar z
6	5	cv 1482	. . . . . . . 8 class z
7		cslt 31794	. . . . . . . 8 class <s
8	4, 6, 7	wbr 4653	. . . . . . 7 wff x <s z
9		vy	. . . . . . . . 9 setvar y
10	9	cv 1482	. . . . . . . 8 class y
11	6, 10, 7	wbr 4653	. . . . . . 7 wff z <s y
12	8, 11	wa 384	. . . . . 6 wff ( x <s z /\ z <s y )
13		cbday 31795	. . . . . . . 8 class bday
14	10, 13	cfv 5888	. . . . . . 7 class ( bday ` y )
15	6, 13	cfv 5888	. . . . . . 7 class ( bday ` z )
16	14, 15	wcel 1990	. . . . . 6 wff ( bday ` y ) e. ( bday ` z )
17	12, 16	wi 4	. . . . 5 wff ( ( x <s z /\ z <s y ) -> ( bday ` y ) e. ( bday ` z ) )
18	17, 5, 3	wral 2912	. . . 4 wff A. z e. No ( ( x <s z /\ z <s y ) -> ( bday ` y ) e. ( bday ` z ) )
19	4, 13	cfv 5888	. . . . 5 class ( bday ` x )
20		cold 31926	. . . . 5 class O
21	19, 20	cfv 5888	. . . 4 class ( O ` ( bday ` x ) )
22	18, 9, 21	crab 2916	. . 3 class { y e. ( O ` ( bday ` x ) ) | A. z e. No ( ( x <s z /\ z <s y ) -> ( bday ` y ) e. ( bday ` z ) ) }
23	2, 3, 22	cmpt 4729	. 2 class ( x e. No |-> { y e. ( O ` ( bday ` x ) ) | A. z e. No ( ( x <s z /\ z <s y ) -> ( bday ` y ) e. ( bday ` z ) ) } )
24	1, 23	wceq 1483	1 wff R = ( x e. No |-> { y e. ( O ` ( bday ` x ) ) | A. z e. No ( ( x <s z /\ z <s y ) -> ( bday ` y ) e. ( bday ` z ) ) } )

This definition is referenced by: (None)