Theorem funbrafv22b

Description: Equivalence of function value and binary relation, analogous to funbrfvb . (Contributed by AV, 6-Sep-2022)

	Ref	Expression
Assertion	funbrafv22b	⊢ ( ( 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		fnbrafv2b	⊢ ( ( 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 ) )