Theorem lsw

Description: Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018)

	Ref	Expression
Assertion	lsw	|- ( W e. X -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elex	|- ( W e. X -> W e. _V )
2		fvex	|- ( W ` ( ( # ` W ) - 1 ) ) e. _V
3		id	|- ( w = W -> w = W )
4		fveq2	|- ( w = W -> ( # ` w ) = ( # ` W ) )
5	4	oveq1d	|- ( w = W -> ( ( # ` w ) - 1 ) = ( ( # ` W ) - 1 ) )
6	3 5	fveq12d	|- ( w = W -> ( w ` ( ( # ` w ) - 1 ) ) = ( W ` ( ( # ` W ) - 1 ) ) )
7		df-lsw	|- lastS = ( w e. _V |-> ( w ` ( ( # ` w ) - 1 ) ) )
8	6 7	fvmptg	|- ( ( W e. _V /\ ( W ` ( ( # ` W ) - 1 ) ) e. _V ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
9	1 2 8	sylancl	|- ( W e. X -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )