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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 )