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Metamath Proof Explorer

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 )