Theorem symg1hash

Description: The symmetric group on a singleton has cardinality 1 . (Contributed by AV, 9-Dec-2018)

	Ref	Expression
Hypotheses	symg1bas.1	|- G = ( SymGrp ` A )
	symg1bas.2	|- B = ( Base ` G )
	symg1bas.0	|- A = { I }
Assertion	symg1hash	|- ( I e. V -> ( # ` B ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		symg1bas.1	|- G = ( SymGrp ` A )
2		symg1bas.2	|- B = ( Base ` G )
3		symg1bas.0	|- A = { I }
4		snfi	|- { I } e. Fin
5	3 4	eqeltri	|- A e. Fin
6	1 2	symghash	|- ( A e. Fin -> ( # ` B ) = ( ! ` ( # ` A ) ) )
7	5 6	ax-mp	|- ( # ` B ) = ( ! ` ( # ` A ) )
8	3	fveq2i	|- ( # ` A ) = ( # ` { I } )
9		hashsng	|- ( I e. V -> ( # ` { I } ) = 1 )
10	8 9	syl5eq	|- ( I e. V -> ( # ` A ) = 1 )
11	10	fveq2d	|- ( I e. V -> ( ! ` ( # ` A ) ) = ( ! ` 1 ) )
12		fac1	|- ( ! ` 1 ) = 1
13	11 12	eqtrdi	|- ( I e. V -> ( ! ` ( # ` A ) ) = 1 )
14	7 13	syl5eq	|- ( I e. V -> ( # ` B ) = 1 )