Definition df-pre 38649

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 38650, dfpre2 38651, dfpre3 38652 and dfpre4 38654.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6259 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6271, cf. also df-bnj14 34845) 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 38636 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38663, which originates from suc11reg 9528. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5840.

Note that dom SucMap = V (see dmsucmap 38642), so the equivalent definition dfpre 38650 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 38380	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1540	. . . 4 class 𝑚
5		csucmap 38378	. . . . . 6 class SucMap
6	5	cdm 5624	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6258	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2113	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6446	. 2 class (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
10	2, 9	wceq 1541	1 wff pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

This definition is referenced by:  dfpre  38650  dfpre4  38654  preex  38665