Theorem dmafv2rnb

Description: The alternate function value at a class A is defined, i.e., in the range of the function, iff A is in the domain of the function. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	dmafv2rnb	⊢ ( Fun ( 𝐹 ↾ { 𝐴 } ) → ( 𝐴 ∈ dom 𝐹 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		iba	⊢ ( Fun ( 𝐹 ↾ { 𝐴 } ) → ( 𝐴 ∈ dom 𝐹 ↔ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) ) )
2		df-dfat	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
3		dfatafv2rnb	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )
4	2 3	bitr3i	⊢ ( ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )
5	1 4	bitrdi	⊢ ( Fun ( 𝐹 ↾ { 𝐴 } ) → ( 𝐴 ∈ dom 𝐹 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 ) )