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Metamath Proof Explorer

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 )