Theorem wlkvtxedg

Description: The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018) (Revised by AV, 2-Jan-2021)

	Ref	Expression
Hypothesis	wlkvtxedg.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	wlkvtxedg	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ 𝐸 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 )

Proof

Step	Hyp	Ref	Expression
1		wlkvtxedg.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
2		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
3	2	wlkvtxiedg	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ ran ( iEdg ‘ 𝐺 ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 )
4		edgval	⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
5	1 4	eqtr2i	⊢ ran ( iEdg ‘ 𝐺 ) = 𝐸
6	5	rexeqi	⊢ ( ∃ 𝑒 ∈ ran ( iEdg ‘ 𝐺 ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 )
7	6	ralbii	⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ ran ( iEdg ‘ 𝐺 ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 ↔ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ 𝐸 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 )
8	3 7	sylib	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∃ 𝑒 ∈ 𝐸 { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ 𝑒 )