Theorem resima

Description: A restriction to an image. (Contributed by NM, 29-Sep-2004)

	Ref	Expression
Assertion	resima	⊢ ( ( 𝐴 ↾ 𝐵 ) “ 𝐵 ) = ( 𝐴 “ 𝐵 )

Proof

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