Theorem imacnvcnv

Description: The image of the double converse of a class. (Contributed by NM, 8-Apr-2007)

	Ref	Expression
Assertion	imacnvcnv	⊢ ( ◡ ◡ 𝐴 “ 𝐵 ) = ( 𝐴 “ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rescnvcnv	⊢ ( ◡ ◡ 𝐴 ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 )
2	1	rneqi	⊢ ran ( ◡ ◡ 𝐴 ↾ 𝐵 ) = ran ( 𝐴 ↾ 𝐵 )
3		df-ima	⊢ ( ◡ ◡ 𝐴 “ 𝐵 ) = ran ( ◡ ◡ 𝐴 ↾ 𝐵 )
4		df-ima	⊢ ( 𝐴 “ 𝐵 ) = ran ( 𝐴 ↾ 𝐵 )
5	2 3 4	3eqtr4i	⊢ ( ◡ ◡ 𝐴 “ 𝐵 ) = ( 𝐴 “ 𝐵 )