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 e. _V
Assertion	fnsnb	|- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )

Proof

Step	Hyp	Ref	Expression
1		fnsnb.1	|- A e. _V
2		fnsnbg	|- ( A e. _V -> ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } ) )
3	1 2	ax-mp	|- ( F Fn { A } <-> F = { <. A , ( F ` A ) >. } )