Definition df-pre 38498

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 38499, dfpre2 38500, dfpre3 38501 and dfpre4 38503.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6248 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6260, cf. also df-bnj14 34701) 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 38485 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38512, which originates from suc11reg 9509. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5830.

Note that dom SucMap = V (see dmsucmap 38491), so the equivalent definition dfpre 38499 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 38229	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1540	. . . 4 class 𝑚
5		csucmap 38227	. . . . . 6 class SucMap
6	5	cdm 5614	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6247	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2111	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6435	. 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  38499  dfpre4  38503  preex  38514