Definition df-pre 39673

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 39674, dfpre2 39675, dfpre3 39676 and dfpre4 39678.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6330 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6342, cf. also df-bnj14 35431) 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 39660 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 39688, which originates from suc11reg 9657. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5885.

Note that dom SucMap = V (see dmsucmap 39666), so the equivalent definition dfpre 39674 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 39378	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1569	. . . 4 class 𝑚
5		csucmap 39376	. . . . . 6 class SucMap
6	5	cdm 5658	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6329	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2147	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6518	. 2 class (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
10	2, 9	wceq 1570	1 wff pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

This definition is used by:  dfpre  39674  dfpre4  39678  preex  39690