Theorem dfatafv2eqfv

Description: If a function is defined at a class A , the alternate function value equals the function's value at A . (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	dfatafv2eqfv	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dfafv22	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , 𝒫 ∪ ran 𝐹 )
2		iftrue	⊢ ( 𝐹 defAt 𝐴 → if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , 𝒫 ∪ ran 𝐹 ) = ( 𝐹 ‘ 𝐴 ) )
3	1 2	syl5eq	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )