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	$$⊢A\in \mathrm{V}$$
Assertion	fnsnb	$$⊢FFn\left\{A\right\}\leftrightarrow F=\left\{⟨A,F\left(A\right)⟩\right\}$$

Proof

Step	Hyp	Ref	Expression
1		fnsnb.1	$$⊢A\in \mathrm{V}$$
2		fnsnbg	$$⊢A\in \mathrm{V}\to \left(FFn\left\{A\right\}\leftrightarrow F=\left\{⟨A,F\left(A\right)⟩\right\}\right)$$
3	1 2	ax-mp	$$⊢FFn\left\{A\right\}\leftrightarrow F=\left\{⟨A,F\left(A\right)⟩\right\}$$