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 ∈ ( 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 ∈ ( 𝑆 MndHom 𝑇 ) ↔ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ) ∧ ( 𝐹 : 𝐵 ⟶ 𝐶 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) ) )
8	7	simprbi	⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → ( 𝐹 : 𝐵 ⟶ 𝐶 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ∧ ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) )
9	8	simp1d	⊢ ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) → 𝐹 : 𝐵 ⟶ 𝐶 )