Definition df-pre 39013

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 39014, dfpre2 39015, dfpre3 39016 and dfpre4 39018.

Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6303 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6315, cf. also df-bnj14 35022) 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 39000 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 39028, which originates from suc11reg 9587. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5882.

Note that dom SucMap = V (see dmsucmap 39006), so the equivalent definition dfpre 39014 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 38718	. 2 class pre 𝑁
3		vm	. . . . 5 setvar 𝑚
4	3	cv 1566	. . . 4 class 𝑚
5		csucmap 38716	. . . . . 6 class SucMap
6	5	cdm 5662	. . . . 5 class dom SucMap
7	6, 5, 1	cpred 6302	. . . 4 class Pred( SucMap , dom SucMap , 𝑁)
8	4, 7	wcel 2149	. . 3 wff 𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁)
9	8, 3	cio 6491	. 2 class (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))
10	2, 9	wceq 1567	1 wff pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

This definition is referenced by:  dfpre  39014  dfpre4  39018  preex  39030