Definition df-pre 39124

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 39125, dfpre2 39126, dfpre3 39127 and dfpre4 39129.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6302 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6314, cf. also df-bnj14 35078) 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 39111 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 39139, which originates from suc11reg 9584. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5881.

Note that dom SucMap = V (see dmsucmap 39117), so the equivalent definition dfpre 39125 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 38829	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1569	. . . 4 class 𝑚
5		csucmap 38827	. . . . . 6 class SucMap
6	5	cdm 5661	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6301	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2143	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6490	. 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 referenced by:  dfpre  39125  dfpre4  39129  preex  39141