Theorem grplactf1o

Description: The left group action of element A of group G maps the underlying set X of G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008) (Proof shortened by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Hypotheses	grplact.1	⊢ 𝐹 = ( 𝑔 ∈ 𝑋 ↦ ( 𝑎 ∈ 𝑋 ↦ ( 𝑔 + 𝑎 ) ) )
	grplact.2	⊢ 𝑋 = ( Base ‘ 𝐺 )
	grplact.3	⊢ + = ( +g ‘ 𝐺 )
Assertion	grplactf1o	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		grplact.1	⊢ 𝐹 = ( 𝑔 ∈ 𝑋 ↦ ( 𝑎 ∈ 𝑋 ↦ ( 𝑔 + 𝑎 ) ) )
2		grplact.2	⊢ 𝑋 = ( Base ‘ 𝐺 )
3		grplact.3	⊢ + = ( +g ‘ 𝐺 )
4		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
5	1 2 3 4	grplactcnv	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 ∧ ◡ ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ ( ( invg ‘ 𝐺 ) ‘ 𝐴 ) ) ) )
6	5	simpld	⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) : 𝑋 –1-1-onto→ 𝑋 )