Definition df-pre 38974

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 38975, dfpre2 38976, dfpre3 38977 and dfpre4 38979.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6288 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6300, cf. also df-bnj14 34985) 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 38961 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 38989, which originates from suc11reg 9574. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5867.

Note that dom SucMap = V (see dmsucmap 38967), so the equivalent definition dfpre 38975 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 38679	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1559	. . . 4 class 𝑚
5		csucmap 38677	. . . . . 6 class SucMap
6	5	cdm 5647	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6287	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2142	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6475	. 2 class (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
10	2, 9	wceq 1560	1 wff pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

This definition is referenced by:  dfpre  38975  dfpre4  38979  preex  38991