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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 )