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Metamath Proof Explorer

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