Definition df-pre 38796

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 38797, dfpre2 38798, dfpre3 38799 and dfpre4 38801.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6265 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6277, cf. also df-bnj14 34832) 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 38783 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38811, which originates from suc11reg 9540. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5846.

Note that dom SucMap = V (see dmsucmap 38789), so the equivalent definition dfpre 38797 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 38501	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1541	. . . 4 class 𝑚
5		csucmap 38499	. . . . . 6 class SucMap
6	5	cdm 5631	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6264	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2114	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6452	. 2 class (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
10	2, 9	wceq 1542	1 wff pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

This definition is referenced by:  dfpre  38797  dfpre4  38801  preex  38813