Definition df-sucmap

Description: Define the successor map, directly as the graph of the successor operation, using only elementary set theory (ordered-pair class abstraction). This avoids committing to any particular construction of the successor function/class from other operators (e.g. a union/composition presentation), while remaining provably equivalent to those presentations (cf. dfsucmap2 and dfsucmap3 vs. df-succf and dfsuccf2 ). For maximum mappy shape, see dfsucmap4 .

We also treat the successor relation as the default shift relation for grading/tower arguments (cf. df-shiftstable ). Because it is used pervasively in shift-lift infrastructure, we adopt the short name SucMap rather than the fully systematic "SucAdjLiftMap".

You may also define the predecessor relation as the converse graph "PreMap" as ` ``' SucMap ` , which reverses successor edges ( cf. cnvopab ) and sends each successor to its (unique) predecessor when it exists. (Contributed by Peter Mazsa, 25-Jan-2026)

	Ref	Expression
Assertion	df-sucmap	Could not format assertion : No typesetting found for |- SucMap = { <. m , n >. | suc m = n } with typecode |-

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		csucmap	Could not format SucMap : No typesetting found for class SucMap with typecode class
1		vm	$$setvarm$$
2		vn	$$setvarn$$
3	1	cv	$$setvarm$$
4	3	csuc	$$class\mathrm{suc}m$$
5	2	cv	$$setvarn$$
6	4 5	wceq	$$wff\mathrm{suc}m=n$$
7	6 1 2	copab	$$class\left\{⟨m,n⟩|\mathrm{suc}m=n\right\}$$
8	0 7	wceq	Could not format SucMap = { <. m , n >. | suc m = n } : No typesetting found for wff SucMap = { <. m , n >. | suc m = n } with typecode wff