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 ∈ 𝑋 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝑊 ∈ 𝑋 → 𝑊 ∈ V )
2		fvex	⊢ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 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 ∈ V ↦ ( 𝑤 ‘ ( ( ♯ ‘ 𝑤 ) − 1 ) ) )
8	6 7	fvmptg	⊢ ( ( 𝑊 ∈ V ∧ ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ V ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
9	1 2 8	sylancl	⊢ ( 𝑊 ∈ 𝑋 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )