Theorem wwlksn0

Description: A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018) (Revised by AV, 9-Apr-2021) (Proof shortened by AV, 21-May-2021)

	Ref	Expression
Hypothesis	wwlkssswrd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	wwlksn0	⊢ ( 𝑊 ∈ ( 0 WWalksN 𝐺 ) → ∃ 𝑣 ∈ 𝑉 𝑊 = ⟨“ 𝑣 ”⟩ )

Proof

Step	Hyp	Ref	Expression
1		wwlkssswrd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		wrdl1exs1	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) → ∃ 𝑣 ∈ ( Vtx ‘ 𝐺 ) 𝑊 = ⟨“ 𝑣 ”⟩ )
3		fveqeq2	⊢ ( 𝑤 = 𝑊 → ( ( ♯ ‘ 𝑤 ) = 1 ↔ ( ♯ ‘ 𝑊 ) = 1 ) )
4		wwlksn0s	⊢ ( 0 WWalksN 𝐺 ) = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 }
5	3 4	elrab2	⊢ ( 𝑊 ∈ ( 0 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) )
6	1	rexeqi	⊢ ( ∃ 𝑣 ∈ 𝑉 𝑊 = ⟨“ 𝑣 ”⟩ ↔ ∃ 𝑣 ∈ ( Vtx ‘ 𝐺 ) 𝑊 = ⟨“ 𝑣 ”⟩ )
7	2 5 6	3imtr4i	⊢ ( 𝑊 ∈ ( 0 WWalksN 𝐺 ) → ∃ 𝑣 ∈ 𝑉 𝑊 = ⟨“ 𝑣 ”⟩ )