Theorem symgbasf

Description: A permutation (element of the symmetric group) is a function from a set into itself. (Contributed by AV, 1-Jan-2019)

	Ref	Expression
Hypotheses	symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
	symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	symgbasf	⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2		symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
3	1 2	symgbasf1o	⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ 𝐴 )
4		f1of	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → 𝐹 : 𝐴 ⟶ 𝐴 )
5	3 4	syl	⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐴 )