Theorem mhmf

Description: A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015)

	Ref	Expression
Hypotheses	mhmf.b	|- B = ( Base ` S )
	mhmf.c	|- C = ( Base ` T )
Assertion	mhmf	|- ( F e. ( S MndHom T ) -> F : B --> C )

Proof

Step	Hyp	Ref	Expression
1		mhmf.b	|- B = ( Base ` S )
2		mhmf.c	|- C = ( Base ` T )
3		eqid	|- ( +g ` S ) = ( +g ` S )
4		eqid	|- ( +g ` T ) = ( +g ` T )
5		eqid	|- ( 0g ` S ) = ( 0g ` S )
6		eqid	|- ( 0g ` T ) = ( 0g ` T )
7	1 2 3 4 5 6	ismhm	|- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) ) )
8	7	simprbi	|- ( F e. ( S MndHom T ) -> ( F : B --> C /\ A. x e. B A. y e. B ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) )
9	8	simp1d	|- ( F e. ( S MndHom T ) -> F : B --> C )