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 )