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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0spth 26101 A pair of an empty set (of edges) and a second set (of vertices) is a simple path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 SPaths 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theorempthistrl 26102 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
(𝐹(𝑉 Paths 𝐸)𝑃𝐹(𝑉 Trails 𝐸)𝑃)

Theoremspthispth 26103 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
(𝐹(𝑉 SPaths 𝐸)𝑃𝐹(𝑉 Paths 𝐸)𝑃)

Theorempthdepisspth 26104 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.)
((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → 𝐹(𝑉 SPaths 𝐸)𝑃)

Theorempthon 26105* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 PathOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})

Theoremispthon 26106 Properties of a pair of functions to be a path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 PathOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Paths 𝐸)𝑃)))

Theorempthonprop 26107 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 PathOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Paths 𝐸)𝑃)))

Theorempthonispth 26108 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 PathOn 𝐸)𝐵)𝑃𝐹(𝑉 Paths 𝐸)𝑃)

Theorem0pthon 26109 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑁𝑉) → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(𝑉 PathOn 𝐸)𝑁)𝑃))

Theorem0pthon1 26110 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑁𝑉) → ∅(𝑁(𝑉 PathOn 𝐸)𝑁){⟨0, 𝑁⟩})

Theorem0pthonv 26111* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝑉𝑋𝐸𝑌) → (𝑁𝑉 → ∃𝑓𝑝 𝑓(𝑁(𝑉 PathOn 𝐸)𝑁)𝑝))

Theoremspthon 26112* The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 SPathOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 SPaths 𝐸)𝑝)})

Theoremisspthon 26113 Properties of a pair of functions to be a simple path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)))

Theoremisspthonpth 26114 Properties of a pair of functions to be a simple path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 9-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝑉 SPaths 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))

Theoremspthonprp 26115 Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
(𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)))

Theoremspthonisspth 26116 A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
(𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)

Theoremspthonepeq 26117 The endpoints of a simple path between two vertices are equal if and only if the path is of length 0 (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
(𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0))

Theoremconstr1trl 26118 Construction of a trail from one given edge in a graph. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
𝐹 = {⟨0, 𝑖⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}       (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐸𝑖) = {𝐴, 𝐵}) → 𝐹(𝑉 Trails 𝐸)𝑃)

Theorem1pthonlem1 26119 Lemma 1 for 1pthon 26121. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
𝐹 = {⟨0, 𝑖⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}       Fun (𝑃 ↾ (1..^(#‘𝐹)))

Theorem1pthonlem2 26120 Lemma 2 for 1pthon 26121. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
𝐹 = {⟨0, 𝑖⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}       ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅

Theorem1pthon 26121 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐸𝑖) = {𝐴, 𝐵}) → {⟨0, 𝑖⟩} (𝐴(𝑉 PathOn 𝐸)𝐵){⟨0, 𝐴⟩, ⟨1, 𝐵⟩})

Theorem1pthoncl 26122 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐼 ∈ V ∧ (𝐸𝐼) = {𝐴, 𝐵})) → {⟨0, 𝐼⟩} (𝐴(𝑉 PathOn 𝐸)𝐵){⟨0, 𝐴⟩, ⟨1, 𝐵⟩})

Theorem1pthon2v 26123* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵}) → ∃𝑓𝑝 𝑓(𝐴(𝑉 PathOn 𝐸)𝐵)𝑝)

Theoremconstr2spthlem1 26124 Lemma 1 for constr2spth 26130. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → Fun 𝑃)

Theorem2pthlem1 26125 Lemma 1 for constr2pth 26131. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       ((𝐴𝑉𝐵𝑉𝐶𝑉) → Fun (𝑃 ↾ (1..^2)))

Theorem2pthlem2 26126 Lemma 2 for constr2pth 26131. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 18-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐵𝐶)) → ((𝑃 “ {0, 2}) ∩ (𝑃 “ (1..^2))) = ∅)

Theorem2wlklem1 26127* Lemma 1 for constr2wlk 26128. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ ((𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶})) → ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})

Theoremconstr2wlk 26128 Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (((𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶}) → 𝐹(𝑉 Walks 𝐸)𝑃))

Theoremconstr2trl 26129 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 1-Feb-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐼𝐽 ∧ (𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶}) → 𝐹(𝑉 Trails 𝐸)𝑃))

Theoremconstr2spth 26130 A simple path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐼𝐽 ∧ (𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶}) → 𝐹(𝑉 SPaths 𝐸)𝑃))

Theoremconstr2pth 26131 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐼𝐽 ∧ (𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶}) → 𝐹(𝑉 Paths 𝐸)𝑃))

Theorem2pthon 26132 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑖𝑗 ∧ (𝐸𝑖) = {𝐴, 𝐵} ∧ (𝐸𝑗) = {𝐵, 𝐶}) → {⟨0, 𝑖⟩, ⟨1, 𝑗⟩} (𝐴(𝑉 PathOn 𝐸)𝐶){⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}))

Theorem2pthoncl 26133 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝐼 ∈ V ∧ 𝐽 ∈ V) ∧ (𝐼𝐽 ∧ (𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶})) → {⟨0, 𝐼⟩, ⟨1, 𝐽⟩} (𝐴(𝑉 PathOn 𝐸)𝐶){⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩})

Theorem2pthon3v 26134* For a vertex adjacent to two other vertices there is a path of length 2 between these other vertices. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ ((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶})) → ∃𝑓𝑝(𝑓(𝐴(𝑉 PathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2))

Theoremredwlklem 26135 Lemma for redwlk 26136. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
((𝐹(𝑉 Walks 𝐸)𝑃 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1))

Theoremredwlk 26136 A walk ending at the last but one vertex of the walk is a walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
((𝐹(𝑉 Walks 𝐸)𝑃 ∧ 1 ≤ (#‘𝐹)) → (𝐹 ↾ (0..^((#‘𝐹) − 1)))(𝑉 Walks 𝐸)(𝑃 ↾ (0..^(#‘𝐹))))

Theoremwlkdvspthlem 26137* Lemma for wlkdvspth 26138. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → Fun 𝐹)

Theoremwlkdvspth 26138 A walk consisting of different vertices is a simple path. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
((𝐹(𝑉 Walks 𝐸)𝑃 ∧ Fun 𝑃) → 𝐹(𝑉 SPaths 𝐸)𝑃)

Theoremusgra2adedgspthlem1 26139 Lemma 1 for usgra2adedgspth 26141. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))

Theoremusgra2adedgspthlem2 26140 Lemma 2 for usgra2adedgspth 26141. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
(((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))

Theoremusgra2adedgspth 26141 In an undirected simple graph, two adjacent edges form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       ((𝑉 USGrph 𝐸𝐴𝐶) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝑉 SPaths 𝐸)𝑃))

Theoremusgra2adedgwlk 26142 In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))))

Theoremusgra2adedgwlkon 26143 In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃))

Theoremusgra2adedgwlkonALT 26144 Alternate proof for usgra2adedgwlkon 26143, using usgra2adedgwlk 26142, but with a longer proof! In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑃))

Theoremusg2wlk 26145* In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))

Theoremusg2wlkon 26146* In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
(𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ∃𝑓𝑝 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝))

Theoremusgra2wlkspthlem1 26147* Lemma 1 for usgra2wlkspth 26149. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
((𝐹 ∈ Word dom 𝐸𝐸:dom 𝐸1-1→ran 𝐸 ∧ (#‘𝐹) = 2) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → Fun 𝐹))

Theoremusgra2wlkspthlem2 26148* Lemma 2 for usgra2wlkspth 26149. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
(((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ (𝑉 USGrph 𝐸𝑃:(0...(#‘𝐹))⟶𝑉)) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → Fun 𝑃))

Theoremusgra2wlkspth 26149 In a undirected simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃))

17.1.5.3  Circuits and cycles

Theoremcrcts 26150* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Circuits 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Theoremcycls 26151* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Cycles 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Paths 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Theoremiscrct 26152 Properties of a pair of functions to be a circuit (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Circuits 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theoremiscycl 26153 Properties of a pair of functions to be a cycle (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theorem0crct 26154 A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 Circuits 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theorem0cycl 26155 A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 Cycles 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theoremcrctistrl 26156 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝐹(𝑉 Circuits 𝐸)𝑃𝐹(𝑉 Trails 𝐸)𝑃)

Theoremcyclispth 26157 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Paths 𝐸)𝑃)

Theoremcycliscrct 26158 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Circuits 𝐸)𝑃)

Theoremcyclnspth 26159 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝐹 ≠ ∅ → (𝐹(𝑉 Cycles 𝐸)𝑃 → ¬ 𝐹(𝑉 SPaths 𝐸)𝑃))

Theoremcycliswlk 26160 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.)
(𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremcyclispthon 26161 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.)
(𝐹(𝑉 Cycles 𝐸)𝑃𝐹((𝑃‘0)(𝑉 PathOn 𝐸)(𝑃‘0))𝑃)

Theoremfargshiftlem 26162 If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
((𝑁 ∈ ℕ0𝑋 ∈ (0..^𝑁)) → (𝑋 + 1) ∈ (1...𝑁))

Theoremfargshiftfv 26163* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (𝑋 ∈ (0..^𝑁) → (𝐺𝑋) = (𝐹‘(𝑋 + 1))))

Theoremfargshiftf 26164* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)

Theoremfargshiftf1 26165* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–1-1→dom 𝐸)

Theoremfargshiftfo 26166* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)–onto→dom 𝐸) → 𝐺:(0..^(#‘𝐹))–onto→dom 𝐸)

Theoremfargshiftfva 26167* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))

Theoremusgrcyclnl1 26168 In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 1)

Theoremusgrcyclnl2 26169 In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 2)

Theorem3cycl3dv 26170 In a simple graph, the vertices of a 3-cycle are mutually different. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐴𝐵𝐵𝐶𝐶𝐴))

Theoremnvnencycllem 26171 Lemma for 3v3e3cycl1 26172 and 4cycl4v4e 26194. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
(((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (𝑋 ∈ ℕ0𝑋 < (#‘𝐹))) → ((𝐸‘(𝐹𝑋)) = {𝐴, 𝐵} → {𝐴, 𝐵} ∈ ran 𝐸))

Theorem3v3e3cycl1 26172* If there is a cycle of length 3 in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((Fun 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))

Theoremconstr3lem1 26173 Lemma for constr3trl 26187 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (𝐹 ∈ V ∧ 𝑃 ∈ V)

Theoremconstr3lem2 26174 Lemma for constr3trl 26187 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (#‘𝐹) = 3

Theoremconstr3lem4 26175 Lemma for constr3trl 26187 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)))

Theoremconstr3lem5 26176 Lemma for constr3trl 26187 etc. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴}))

Theoremconstr3lem6 26177 Lemma for constr3pthlem3 26185. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({(𝑃‘0), (𝑃‘3)} ∩ {(𝑃‘1), (𝑃‘2)}) = ∅)

Theoremconstr3trllem1 26178 Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 ∈ Word dom 𝐸)

Theoremconstr3trllem2 26179 Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → Fun 𝐹)

Theoremconstr3trllem3 26180 Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝑃:(0...(#‘𝐹))⟶𝑉)

Theoremconstr3trllem4 26181 Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝑃:(0...3)⟶𝑉)

Theoremconstr3trllem5 26182* Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})

Theoremconstr3pthlem1 26183 Lemma for constr3pth 26188. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐵𝑉𝐶𝑊) → (𝑃 ↾ (1..^(#‘𝐹))) = {⟨1, 𝐵⟩, ⟨2, 𝐶⟩})

Theoremconstr3pthlem2 26184 Lemma for constr3pth 26188. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐵𝐶) → Fun (𝑃 ↾ (1..^(#‘𝐹))))

Theoremconstr3pthlem3 26185 Lemma for constr3pth 26188. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)

Theoremconstr3cycllem1 26186 Lemma for constr3cycl 26189. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑃‘0) = (𝑃‘(#‘𝐹)))

Theoremconstr3trl 26187 Construction of a trail from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹(𝑉 Trails 𝐸)𝑃)

Theoremconstr3pth 26188 Construction of a path from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹(𝑉 Paths 𝐸)𝑃)

Theoremconstr3cycl 26189 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3))

Theoremconstr3cyclp 26190 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}    &   𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})       ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴))

Theoremconstr3cyclpe 26191* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

Theorem3v3e3cycl2 26192* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
(𝑉 USGrph 𝐸 → (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → ∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3)))

Theorem3v3e3cycl 26193* If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
(𝑉 USGrph 𝐸 → (∃𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))

Theorem4cycl4v4e 26194* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((Fun 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))

Theorem4cycl4dv 26195 In a simple graph, the vertices of a 4-cycle are mutually different. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹 ∧ (#‘𝐹) = 4)) → ((((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}) ∧ ((𝐸‘(𝐹‘2)) = {𝐶, 𝐷} ∧ (𝐸‘(𝐹‘3)) = {𝐷, 𝐴})) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) ∧ ((𝐴𝐵𝐴𝐶𝐴𝐷) ∧ (𝐵𝐶𝐵𝐷𝐶𝐷)))))

Theorem4cycl4dv4e 26196* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))))

17.1.5.4  Connected graphs

Syntaxcconngra 26197 Extend class notation with connected graphs.
class ConnGrph

Definitiondf-conngra 26198* Define the class of all connected graphs. A graph (or, more generally, any pair representing a structure consisting of "vertices" and "edges") is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv 26111 and dfconngra1 26199. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
ConnGrph = {⟨𝑣, 𝑒⟩ ∣ ∀𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝}

Theoremdfconngra1 26199* Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
ConnGrph = {⟨𝑣, 𝑒⟩ ∣ ∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝}

Theoremisconngra 26200* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))

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