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 e. Fin -> B e. Fin )

Proof

Step	Hyp	Ref	Expression
1		symgbas.1	|- G = ( SymGrp ` A )
2		symgbas.2	|- B = ( Base ` G )
3		mapfi	|- ( ( A e. Fin /\ A e. Fin ) -> ( A ^m A ) e. Fin )
4	3	anidms	|- ( A e. Fin -> ( A ^m A ) e. 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 } C_ { f | f : A --> A }
8	5 7	eqsstri	|- B C_ { f | f : A --> A }
9		mapvalg	|- ( ( A e. Fin /\ A e. Fin ) -> ( A ^m A ) = { f | f : A --> A } )
10	9	anidms	|- ( A e. Fin -> ( A ^m A ) = { f | f : A --> A } )
11	8 10	sseqtrrid	|- ( A e. Fin -> B C_ ( A ^m A ) )
12	4 11	ssfid	|- ( A e. Fin -> B e. Fin )