This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )