Theorem mptima2

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

	Ref	Expression
Hypothesis	mptima2.1	|- ( ph -> C C_ A )
Assertion	mptima2	|- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) )

Proof

Step	Hyp	Ref	Expression
1		mptima2.1	|- ( ph -> C C_ A )
2		mptima	|- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )
3		sseqin2	|- ( C C_ A <-> ( A i^i C ) = C )
4	1 3	sylib	|- ( ph -> ( A i^i C ) = C )
5	4	mpteq1d	|- ( ph -> ( x e. ( A i^i C ) |-> B ) = ( x e. C |-> B ) )
6	5	rneqd	|- ( ph -> ran ( x e. ( A i^i C ) |-> B ) = ran ( x e. C |-> B ) )
7	2 6	syl5eq	|- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) )