Definition df-pre 39224

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 39225, dfpre2 39226, dfpre3 39227 and dfpre4 39229.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6299 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6311, cf. also df-bnj14 35200) 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 39211 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 39239, which originates from suc11reg 9599. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5875.

Note that dom SucMap = V (see dmsucmap 39217), so the equivalent definition dfpre 39225 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 38929	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1569	. . . 4 class 𝑚
5		csucmap 38927	. . . . . 6 class SucMap
6	5	cdm 5655	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6298	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2145	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6487	. 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  39225  dfpre4  39229  preex  39241