Theorem ghmf

Description: A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014)

	Ref	Expression
Hypotheses	ghmf.x	⊢ 𝑋 = ( Base ‘ 𝑆 )
	ghmf.y	⊢ 𝑌 = ( Base ‘ 𝑇 )
Assertion	ghmf	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 : 𝑋 ⟶ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		ghmf.x	⊢ 𝑋 = ( Base ‘ 𝑆 )
2		ghmf.y	⊢ 𝑌 = ( Base ‘ 𝑇 )
3		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
4		eqid	⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 )
5	1 2 3 4	isghm	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ↔ ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝐹 ‘ ( 𝑦 ( +g ‘ 𝑆 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) ) ) )
6	5	simprbi	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑥 ∈ 𝑋 ( 𝐹 ‘ ( 𝑦 ( +g ‘ 𝑆 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) ) )
7	6	simpld	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 : 𝑋 ⟶ 𝑌 )