Theorem ghmima

Description: The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014)

	Ref	Expression
Assertion	ghmima	|- ( ( F e. ( S GrpHom T ) /\ U e. ( SubGrp ` S ) ) -> ( F " U ) e. ( SubGrp ` T ) )

Proof

Step	Hyp	Ref	Expression
1		df-ima	|- ( F " U ) = ran ( F |` U )
2		eqid	|- ( S |`s U ) = ( S |`s U )
3	2	resghm	|- ( ( F e. ( S GrpHom T ) /\ U e. ( SubGrp ` S ) ) -> ( F |` U ) e. ( ( S |`s U ) GrpHom T ) )
4		ghmrn	|- ( ( F |` U ) e. ( ( S |`s U ) GrpHom T ) -> ran ( F |` U ) e. ( SubGrp ` T ) )
5	3 4	syl	|- ( ( F e. ( S GrpHom T ) /\ U e. ( SubGrp ` S ) ) -> ran ( F |` U ) e. ( SubGrp ` T ) )
6	1 5	eqeltrid	|- ( ( F e. ( S GrpHom T ) /\ U e. ( SubGrp ` S ) ) -> ( F " U ) e. ( SubGrp ` T ) )