Theorem mgmhmf

Description: A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020)

	Ref	Expression
Hypotheses	mgmhmf.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	mgmhmf.c	⊢ 𝐶 = ( Base ‘ 𝑇 )
Assertion	mgmhmf	⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → 𝐹 : 𝐵 ⟶ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		mgmhmf.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
2		mgmhmf.c	⊢ 𝐶 = ( Base ‘ 𝑇 )
3		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
4		eqid	⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 )
5	1 2 3 4	ismgmhm	⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ↔ ( ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ∧ ( 𝐹 : 𝐵 ⟶ 𝐶 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) ) )
6		simprl	⊢ ( ( ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ∧ ( 𝐹 : 𝐵 ⟶ 𝐶 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ) ) → 𝐹 : 𝐵 ⟶ 𝐶 )
7	5 6	sylbi	⊢ ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) → 𝐹 : 𝐵 ⟶ 𝐶 )