Theorem mptima

Description: Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Assertion	mptima	|- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )

Proof

Step	Hyp	Ref	Expression
1		df-ima	|- ( ( x e. A |-> B ) " C ) = ran ( ( x e. A |-> B ) |` C )
2		resmpt3	|- ( ( x e. A |-> B ) |` C ) = ( x e. ( A i^i C ) |-> B )
3	2	rneqi	|- ran ( ( x e. A |-> B ) |` C ) = ran ( x e. ( A i^i C ) |-> B )
4	1 3	eqtri	|- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )