Theorem funbrfvb

Description: Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006)

	Ref	Expression
Assertion	funbrfvb	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐴 ) = 𝐵 ↔ 𝐴 𝐹 𝐵 ) )

Proof

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