Theorem aovvfunressn

Description: If the operation value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	aovvfunressn	⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝐶 → Fun ( 𝐹 ↾ { ⟨ 𝐴 , 𝐵 ⟩ } ) )

Proof

Step	Hyp	Ref	Expression
1		df-aov	⊢ (( 𝐴 𝐹 𝐵 )) = ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ )
2	1	eleq1i	⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝐶 ↔ ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ ) ∈ 𝐶 )
3		afvvfunressn	⊢ ( ( 𝐹 ''' ⟨ 𝐴 , 𝐵 ⟩ ) ∈ 𝐶 → Fun ( 𝐹 ↾ { ⟨ 𝐴 , 𝐵 ⟩ } ) )
4	2 3	sylbi	⊢ ( (( 𝐴 𝐹 𝐵 )) ∈ 𝐶 → Fun ( 𝐹 ↾ { ⟨ 𝐴 , 𝐵 ⟩ } ) )