Theorem elmptima

Description: The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Assertion	elmptima	|- ( C e. V -> ( C e. ( ( x e. A |-> B ) " D ) <-> E. x e. ( A i^i D ) C = B ) )

Proof

Step	Hyp	Ref	Expression
1		mptima	|- ( ( x e. A |-> B ) " D ) = ran ( x e. ( A i^i D ) |-> B )
2	1	a1i	|- ( C e. V -> ( ( x e. A |-> B ) " D ) = ran ( x e. ( A i^i D ) |-> B ) )
3	2	eleq2d	|- ( C e. V -> ( C e. ( ( x e. A |-> B ) " D ) <-> C e. ran ( x e. ( A i^i D ) |-> B ) ) )
4		eqid	|- ( x e. ( A i^i D ) |-> B ) = ( x e. ( A i^i D ) |-> B )
5	4	elrnmpt	|- ( C e. V -> ( C e. ran ( x e. ( A i^i D ) |-> B ) <-> E. x e. ( A i^i D ) C = B ) )
6	3 5	bitrd	|- ( C e. V -> ( C e. ( ( x e. A |-> B ) " D ) <-> E. x e. ( A i^i D ) C = B ) )