Definition df-pre 38588

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 38589, dfpre2 38590, dfpre3 38591 and dfpre4 38593.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6257 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6269, cf. also df-bnj14 34794) 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 38575 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38602, which originates from suc11reg 9526. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5838.

Note that dom SucMap = V (see dmsucmap 38581), so the equivalent definition dfpre 38589 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 38319	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1540	. . . 4 class 𝑚
5		csucmap 38317	. . . . . 6 class SucMap
6	5	cdm 5622	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6256	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2113	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6444	. 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  38589  dfpre4  38593  preex  38604