Definition df-pre 39544

Description: Define the term-level successor-predecessor. It is the unique 𝑚 with suc 𝑚 = 𝑁 when such an 𝑚 exists; otherwise pre 𝑁 is the arbitrary default chosen by ℩. See its alternate definitions dfpre 39545, dfpre2 39546, dfpre3 39547 and dfpre4 39549.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6307 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6319, cf. also df-bnj14 35370) in several respects. Its most abstract property as a specialisation is that it has a unique existing value by default. This is in contrast to the general version. The uniqueness (conditional on existence) is implied by the property of this specific instance of the general case involving the successor map df-sucmap 39531 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 39559, which originates from suc11reg 9627. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5874.

Note that dom SucMap = V (see dmsucmap 39537), so the equivalent definition dfpre 39545 uses (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁)). (Contributed by Peter Mazsa, 27-Jan-2026.)

Assertion

Ref	Expression
df-pre	⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

Distinct variable group:   𝑚,𝑁

Detailed syntax breakdown of Definition df-pre

Step	Hyp	Ref	Expression
1		cN	. . 3 class 𝑁
2	1	cpre 39249	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1568	. . . 4 class 𝑚
5		csucmap 39247	. . . . . 6 class SucMap
6	5	cdm 5651	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6306	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2144	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6495	. 2 class (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
10	2, 9	wceq 1569	1 wff pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

This definition is used by:  dfpre  39545  dfpre4  39549  preex  39561