Definition df-pre 38813

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 38814, dfpre2 38815, dfpre3 38816 and dfpre4 38818.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6260 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6272, cf. also df-bnj14 34851) 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 38800 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38828, which originates from suc11reg 9534. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5841.

Note that dom SucMap = V (see dmsucmap 38806), so the equivalent definition dfpre 38814 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 38518	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1541	. . . 4 class 𝑚
5		csucmap 38516	. . . . . 6 class SucMap
6	5	cdm 5625	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6259	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2114	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6447	. 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  38814  dfpre4  38818  preex  38830