Theorem dfafv22

Description: Alternate definition of ( F '''' A ) using ( FA ) directly. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	dfafv22	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , 𝒫 ∪ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		df-afv2	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 )
2		df-fv	⊢ ( 𝐹 ‘ 𝐴 ) = ( ℩ 𝑥 𝐴 𝐹 𝑥 )
3	2	eqcomi	⊢ ( ℩ 𝑥 𝐴 𝐹 𝑥 ) = ( 𝐹 ‘ 𝐴 )
4		ifeq1	⊢ ( ( ℩ 𝑥 𝐴 𝐹 𝑥 ) = ( 𝐹 ‘ 𝐴 ) → if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 ) = if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , 𝒫 ∪ ran 𝐹 ) )
5	3 4	ax-mp	⊢ if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 ) = if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , 𝒫 ∪ ran 𝐹 )
6	1 5	eqtri	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( 𝐹 ‘ 𝐴 ) , 𝒫 ∪ ran 𝐹 )