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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 ∈ 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