Theorem dfatafv2iota

Description: If a function is defined at a class A the alternate function value at A is the unique value assigned to A by the function (analogously to ( FA ) ). (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	dfatafv2iota	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = ( ℩ 𝑥 𝐴 𝐹 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		df-afv2	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 )
2		iftrue	⊢ ( 𝐹 defAt 𝐴 → if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 ) = ( ℩ 𝑥 𝐴 𝐹 𝑥 ) )
3	1 2	syl5eq	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = ( ℩ 𝑥 𝐴 𝐹 𝑥 ) )