Theorem funopfvb

Description: Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of Monk1 p. 42. (Contributed by NM, 26-Jan-1997)

	Ref	Expression
Assertion	funopfvb	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐴 ) = 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		funfn	⊢ ( Fun 𝐹 ↔ 𝐹 Fn dom 𝐹 )
2		fnopfvb	⊢ ( ( 𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐴 ) = 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐹 ) )
3	1 2	sylanb	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐴 ) = 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐹 ) )