Definition df-pre 38726

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 38727, dfpre2 38728, dfpre3 38729 and dfpre4 38731.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6267 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6279, cf. also df-bnj14 34866) 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 38713 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38741, which originates from suc11reg 9540. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5848.

Note that dom SucMap = V (see dmsucmap 38719), so the equivalent definition dfpre 38727 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 38431	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1541	. . . 4 class 𝑚
5		csucmap 38429	. . . . . 6 class SucMap
6	5	cdm 5632	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6266	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2114	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6454	. 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  38727  dfpre4  38731  preex  38743