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Metamath Proof Explorer

Theorem symgbasf1o

Description: Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018)

Ref Expression
Hypotheses symgbas.1 ⊢ G = SymGrp ⁡ A
symgbas.2 ⊢ B = Base G
Assertion symgbasf1o ⊢ F ∈ B → F : A ⟶ 1-1 onto A

Proof

Step Hyp Ref Expression
1 symgbas.1 ⊢ G = SymGrp ⁡ A
2 symgbas.2 ⊢ B = Base G
3 1 2 elsymgbas2 ⊢ F ∈ B → F ∈ B ↔ F : A ⟶ 1-1 onto A
4 3 ibi ⊢ F ∈ B → F : A ⟶ 1-1 onto A