Theorem fnsnb

Description: A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) Revised to add reverse implication. (Revised by NM, 29-Dec-2018) (Proof shortened by Zhi Wang, 21-Oct-2025)

	Ref	Expression
Hypothesis	fnsnb.1	⊢ 𝐴 ∈ V
Assertion	fnsnb	⊢ ( 𝐹 Fn { 𝐴 } ↔ 𝐹 = { ⟨ 𝐴 , ( 𝐹 ‘ 𝐴 ) ⟩ } )

Proof

Step	Hyp	Ref	Expression
1		fnsnb.1	⊢ 𝐴 ∈ V
2		fnsnbg	⊢ ( 𝐴 ∈ V → ( 𝐹 Fn { 𝐴 } ↔ 𝐹 = { ⟨ 𝐴 , ( 𝐹 ‘ 𝐴 ) ⟩ } ) )
3	1 2	ax-mp	⊢ ( 𝐹 Fn { 𝐴 } ↔ 𝐹 = { ⟨ 𝐴 , ( 𝐹 ‘ 𝐴 ) ⟩ } )