Definition df-pre 39407

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 39408, dfpre2 39409, dfpre3 39410 and dfpre4 39412.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6304 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6316, cf. also df-bnj14 35320) 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 39394 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 39422, which originates from suc11reg 9620. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5873.

Note that dom SucMap = V (see dmsucmap 39400), so the equivalent definition dfpre 39408 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 39112	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1569	. . . 4 class 𝑚
5		csucmap 39110	. . . . . 6 class SucMap
6	5	cdm 5651	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6303	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2145	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6492	. 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  39408  dfpre4  39412  preex  39424