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Metamath Proof Explorer

Theorem symgbasfi

Description: The symmetric group on a finite index set is finite. (Contributed by SO, 9-Jul-2018)

Ref Expression
Hypotheses symgbas.1 ⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
symgbas.2 ⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion symgbasfi ( 𝐴 ∈ Fin → 𝐵 ∈ Fin )

Proof

Step Hyp Ref Expression
1 symgbas.1 ⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2 symgbas.2 ⊢ 𝐵 = ( Base ‘ 𝐺 )
3 mapfi ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ∈ Fin ) → ( 𝐴 ↑m 𝐴 ) ∈ Fin )
4 3 anidms ⊢ ( 𝐴 ∈ Fin → ( 𝐴 ↑m 𝐴 ) ∈ Fin )
5 1 2 symgbas ⊢ 𝐵 = { 𝑓 ∣ 𝑓 : 𝐴 –1-1-onto→ 𝐴 }
6 f1of ⊢ ( 𝑓 : 𝐴 –1-1-onto→ 𝐴 → 𝑓 : 𝐴 ⟶ 𝐴 )
7 6 ss2abi ⊢ { 𝑓 ∣ 𝑓 : 𝐴 –1-1-onto→ 𝐴 } ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 }
8 5 7 eqsstri ⊢ 𝐵 ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 }
9 mapvalg ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ∈ Fin ) → ( 𝐴 ↑m 𝐴 ) = { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } )
10 9 anidms ⊢ ( 𝐴 ∈ Fin → ( 𝐴 ↑m 𝐴 ) = { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } )
11 8 10 sseqtrrid ⊢ ( 𝐴 ∈ Fin → 𝐵 ⊆ ( 𝐴 ↑m 𝐴 ) )
12 4 11 ssfid ⊢ ( 𝐴 ∈ Fin → 𝐵 ∈ Fin )