Theorem dfatopafv2b

Description: Equivalence of function value and ordered pair membership, analogous to fnopfvb or funopfvb . (Contributed by AV, 6-Sep-2022)

	Ref	Expression
Assertion	dfatopafv2b	⊢ ( ( 𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐹 '''' 𝐴 ) = 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		dfatbrafv2b	⊢ ( ( 𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐹 '''' 𝐴 ) = 𝐵 ↔ 𝐴 𝐹 𝐵 ) )
2		df-br	⊢ ( 𝐴 𝐹 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐹 )
3	1 2	bitrdi	⊢ ( ( 𝐹 defAt 𝐴 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝐹 '''' 𝐴 ) = 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝐹 ) )