Theorem fundmafv2rnb

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	fundmafv2rnb	⊢ ( Fun 𝐹 → ( 𝐴 ∈ dom 𝐹 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		funres	⊢ ( Fun 𝐹 → Fun ( 𝐹 ↾ { 𝐴 } ) )
2		dmafv2rnb	⊢ ( Fun ( 𝐹 ↾ { 𝐴 } ) → ( 𝐴 ∈ dom 𝐹 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 ) )
3	1 2	syl	⊢ ( Fun 𝐹 → ( 𝐴 ∈ dom 𝐹 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 ) )