Theorem ovn0dmfun

Description: If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 . (Contributed by AV, 27-Jan-2020)

	Ref	Expression
Assertion	ovn0dmfun	⊢ ( ( 𝐴 𝐹 𝐵 ) ≠ ∅ → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-ov	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ )
2	1	neeq1i	⊢ ( ( 𝐴 𝐹 𝐵 ) ≠ ∅ ↔ ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ≠ ∅ )
3		fvfundmfvn0	⊢ ( ( 𝐹 ‘ ⟨ 𝐴 , 𝐵 ⟩ ) ≠ ∅ → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) )
4	2 3	sylbi	⊢ ( ( 𝐴 𝐹 𝐵 ) ≠ ∅ → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { ⟨ 𝐴 , 𝐵 ⟩ } ) ) )