Theorem afv2rnfveq

Description: If the alternate function value is defined, i.e., in the range of the function, the alternate function value equals the function's value. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv2rnfveq	⊢ ( ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dfatafv2rnb	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )
2		dfatafv2eqfv	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
3	1 2	sylbir	⊢ ( ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )