Theorem dfpre

Description: Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 27-Jan-2026)

	Ref	Expression
Assertion	dfpre	⊢ pre 𝑁 = ( ℩ 𝑚 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		df-pre	⊢ pre 𝑁 = ( ℩ 𝑚 𝑚 ∈ Pred ( SucMap , dom SucMap , 𝑁 ) )
2		dmsucmap	⊢ dom SucMap = V
3		predeq2	⊢ ( dom SucMap = V → Pred ( SucMap , dom SucMap , 𝑁 ) = Pred ( SucMap , V , 𝑁 ) )
4	2 3	ax-mp	⊢ Pred ( SucMap , dom SucMap , 𝑁 ) = Pred ( SucMap , V , 𝑁 )
5	4	eleq2i	⊢ ( 𝑚 ∈ Pred ( SucMap , dom SucMap , 𝑁 ) ↔ 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) )
6	5	iotabii	⊢ ( ℩ 𝑚 𝑚 ∈ Pred ( SucMap , dom SucMap , 𝑁 ) ) = ( ℩ 𝑚 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) )
7	1 6	eqtri	⊢ pre 𝑁 = ( ℩ 𝑚 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) )