Theorem imadmres

Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007)

	Ref	Expression
Assertion	imadmres	⊢ ( 𝐴 “ dom ( 𝐴 ↾ 𝐵 ) ) = ( 𝐴 “ 𝐵 )

Proof

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