Theorem dfpre3

Description: Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026)

	Ref	Expression
Assertion	dfpre3	|- ( N e. V -> pre N = ( iota m suc m = N ) )

Proof

Step	Hyp	Ref	Expression
1		dfpre2	|- ( N e. V -> pre N = ( iota m m SucMap N ) )
2		brsucmap	|- ( ( m e. _V /\ N e. V ) -> ( m SucMap N <-> suc m = N ) )
3	2	el2v1	|- ( N e. V -> ( m SucMap N <-> suc m = N ) )
4	3	iotabidv	|- ( N e. V -> ( iota m m SucMap N ) = ( iota m suc m = N ) )
5	1 4	eqtrd	|- ( N e. V -> pre N = ( iota m suc m = N ) )