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Metamath Proof Explorer

Theorem wwlknon

Description: An element of the set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018) (Revised by AV, 12-May-2021) (Revised by AV, 14-Mar-2022)

Ref Expression
Assertion wwlknon ( 𝑊 ∈ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ↔ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( 𝑊 ‘ 0 ) = 𝐴 ∧ ( 𝑊 ‘ 𝑁 ) = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 fveq1 ⊢ ( 𝑤 = 𝑊 → ( 𝑤 ‘ 0 ) = ( 𝑊 ‘ 0 ) )
2 1 eqeq1d ⊢ ( 𝑤 = 𝑊 → ( ( 𝑤 ‘ 0 ) = 𝐴 ↔ ( 𝑊 ‘ 0 ) = 𝐴 ) )
3 fveq1 ⊢ ( 𝑤 = 𝑊 → ( 𝑤 ‘ 𝑁 ) = ( 𝑊 ‘ 𝑁 ) )
4 3 eqeq1d ⊢ ( 𝑤 = 𝑊 → ( ( 𝑤 ‘ 𝑁 ) = 𝐵 ↔ ( 𝑊 ‘ 𝑁 ) = 𝐵 ) )
5 2 4 anbi12d ⊢ ( 𝑤 = 𝑊 → ( ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) ↔ ( ( 𝑊 ‘ 0 ) = 𝐴 ∧ ( 𝑊 ‘ 𝑁 ) = 𝐵 ) ) )
6 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
7 6 iswwlksnon ⊢ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) = { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) }
8 5 7 elrab2 ⊢ ( 𝑊 ∈ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ↔ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( ( 𝑊 ‘ 0 ) = 𝐴 ∧ ( 𝑊 ‘ 𝑁 ) = 𝐵 ) ) )
9 3anass ⊢ ( ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( 𝑊 ‘ 0 ) = 𝐴 ∧ ( 𝑊 ‘ 𝑁 ) = 𝐵 ) ↔ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( ( 𝑊 ‘ 0 ) = 𝐴 ∧ ( 𝑊 ‘ 𝑁 ) = 𝐵 ) ) )
10 8 9 bitr4i ⊢ ( 𝑊 ∈ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ↔ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( 𝑊 ‘ 0 ) = 𝐴 ∧ ( 𝑊 ‘ 𝑁 ) = 𝐵 ) )