Definition df-left 27570

Description: Define the left options of a surreal. This is the set of surreals that are simpler and less than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.)

Assertion

Ref	Expression
df-left	⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-left

Step	Hyp	Ref	Expression
1		cleft 27565	. 2 class L
2		vx	. . 3 setvar 𝑥
3		csur 27367	. . 3 class No
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1540	. . . . 5 class 𝑦
6	2	cv 1540	. . . . 5 class 𝑥
7		cslt 27368	. . . . 5 class <s
8	5, 6, 7	wbr 5148	. . . 4 wff 𝑦 <s 𝑥
9		cbday 27369	. . . . . 6 class bday
10	6, 9	cfv 6543	. . . . 5 class ( bday ‘𝑥)
11		cold 27563	. . . . 5 class O
12	10, 11	cfv 6543	. . . 4 class ( O ‘( bday ‘𝑥))
13	8, 4, 12	crab 3432	. . 3 class {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥}
14	2, 3, 13	cmpt 5231	. 2 class (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
15	1, 14	wceq 1541	1 wff L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})

This definition is referenced by:  leftval  27583  leftf  27585