Theorem dfpre2

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

	Ref	Expression
Assertion	dfpre2	⊢ ( 𝑁 ∈ 𝑉 → pre 𝑁 = ( ℩ 𝑚 𝑚 SucMap 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		dfpre	⊢ pre 𝑁 = ( ℩ 𝑚 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) )
2		elpredg	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑚 ∈ V ) → ( 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) ↔ 𝑚 SucMap 𝑁 ) )
3	2	elvd	⊢ ( 𝑁 ∈ 𝑉 → ( 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) ↔ 𝑚 SucMap 𝑁 ) )
4	3	iotabidv	⊢ ( 𝑁 ∈ 𝑉 → ( ℩ 𝑚 𝑚 ∈ Pred ( SucMap , V , 𝑁 ) ) = ( ℩ 𝑚 𝑚 SucMap 𝑁 ) )
5	1 4	eqtrid	⊢ ( 𝑁 ∈ 𝑉 → pre 𝑁 = ( ℩ 𝑚 𝑚 SucMap 𝑁 ) )