Theorem eupthseg

Description: The N -th edge in an eulerian path is the edge having P ( N ) and P ( N + 1 ) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 18-Feb-2021)

	Ref	Expression
Hypothesis	eupths.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	eupthseg	⊢ ( ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eupths.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2	1	eupthi	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1-onto→ dom 𝐼 ) )
3	2	simpld	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
4	1	wlkvtxeledg	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )
5		fveq2	⊢ ( 𝑘 = 𝑁 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 𝑁 ) )
6		fvoveq1	⊢ ( 𝑘 = 𝑁 → ( 𝑃 ‘ ( 𝑘 + 1 ) ) = ( 𝑃 ‘ ( 𝑁 + 1 ) ) )
7	5 6	preq12d	⊢ ( 𝑘 = 𝑁 → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } )
8		2fveq3	⊢ ( 𝑘 = 𝑁 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) )
9	7 8	sseq12d	⊢ ( 𝑘 = 𝑁 → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
10	9	rspccv	⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
11	3 4 10	3syl	⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) ) )
12	11	imp	⊢ ( ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → { ( 𝑃 ‘ 𝑁 ) , ( 𝑃 ‘ ( 𝑁 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑁 ) ) )