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Metamath Proof Explorer

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 ) ) )