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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwlkonprop 25801 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))

Theoremwlkoniswlk 25802 A walk between to vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremwlkonwlk 25803 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
(𝐹(𝑉 Walks 𝐸)𝑃𝐹((𝑃‘0)(𝑉 WalkOn 𝐸)(𝑃‘(#‘𝐹)))𝑃)

Theoremtrls 25804* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Trails 𝐸) = {⟨𝑓, 𝑝⟩ ∣ ((𝑓 ∈ Word dom 𝐸 ∧ Fun 𝑓) ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Theoremistrl 25805* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremistrl2 25806* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremtrliswlk 25807 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(𝐹(𝑉 Trails 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremtrlon 25808* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 TrailOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Trails 𝐸)𝑝)})

Theoremistrlon 25809 Properties of a pair of functions to be a trail between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Trails 𝐸)𝑃)))

Theoremtrlonprop 25810 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
(𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Trails 𝐸)𝑃)))

Theoremtrlonistrl 25811 A trail between to vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃𝐹(𝑉 Trails 𝐸)𝑃)

Theoremtrlonwlkon 25812 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
(𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃)

Theorem0wlk 25813 A pair of an empty set (of edges) and a second set (of vertices) is a walk 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 Walks 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theorem0trl 25814 A pair of an empty set (of edges) and a second set (of vertices) is a trail if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Proof shortened by AV, 7-Jan-2020.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 Trails 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theorem0wlkon 25815 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑁𝑉) → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(𝑉 WalkOn 𝐸)𝑁)𝑃))

Theorem0trlon 25816 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑁𝑉) → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(𝑉 TrailOn 𝐸)𝑁)𝑃))

Theorem2trllemF 25817 Lemma 5 for constr2trl 25867. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(((𝐸𝐼) = {𝑋, 𝑌} ∧ 𝑌𝑉) → 𝐼 ∈ dom 𝐸)

Theorem2trllemA 25818 Lemma 1 for constr2trl 25867. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (#‘𝐹) = 2

Theorem2trllemB 25819 Lemma 2 for constr2trl 25867. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (0..^(#‘𝐹)) = {0, 1}

Theorem2trllemH 25820 Lemma 3 for constr2trl 25867. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (((𝑉𝑋𝐸𝑌𝐵𝑉) ∧ ((𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)

Theorem2trllemE 25821 Lemma 4 for constr2trl 25867. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (((𝑉𝑋𝐸𝑌𝐵𝑉) ∧ 𝐼𝐽 ∧ ((𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸)

Theorem2wlklemA 25822 Lemma for constr2wlk 25866. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝐴𝑉 → (𝑃‘0) = 𝐴)

Theorem2wlklemB 25823 Lemma for constr2wlk 25866. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝐵𝑉 → (𝑃‘1) = 𝐵)

Theorem2wlklemC 25824 Lemma for constr2wlk 25866. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝐶𝑉 → (𝑃‘2) = 𝐶)

Theorem2trllemD 25825 Lemma 4 for constr2trl 25867. (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) → 𝑃 Fn {0, 1, 2})

Theorem2trllemG 25826 Lemma 7 for constr2trl 25867. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝑃:(0...2)⟶𝑉)

Theoremwlkntrllem1 25827 Lemma 1 for wlkntrl 25830: F is a word over {0}, the domain of E. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}       𝐹 ∈ Word dom 𝐸

Theoremwlkntrllem2 25828* Lemma 2 for wlkntrl 25830: The values of E after F are edges between two vertices enumerated by P. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}       𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}

Theoremwlkntrllem3 25829* Lemma 3 for wlkntrl 25830: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}        ¬ Fun 𝐹

Theoremwlkntrl 25830* A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one edge to the other is a walk, but not a trail. Notice that 𝑉, 𝐸 is a simple graph (without loops) only if 𝑥𝑦. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}       (𝐹(𝑉 Walks 𝐸)𝑃 ∧ ¬ 𝐹(𝑉 Trails 𝐸)𝑃)

Theoremusgrwlknloop 25831* In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Walks 𝐸)𝑃) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

Theorem2wlklem 25832* Lemma for is2wlk 25833 and 2wlklemA 25822. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
(∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))

Theoremis2wlk 25833 Properties of a pair of functions to be a walk of length 2 (in an undirected graph). (Contributed by Alexander van der Vekens, 16-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐸𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

16.1.5.2  Paths and simple paths

Theorempths 25834* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Paths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})

Theoremspths 25835* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 SPaths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun 𝑝)})

Theoremispth 25836 Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))

Theoremisspth 25837 Properties of a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))

Theorem0pth 25838 A pair of an empty set (of edges) and a second set (of vertices) is a 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 Paths 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theorem0spth 25839 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 25840 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 25841 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 25842 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 25843* 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 25844 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 25845 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 25846 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 25847 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 25848 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 25849* 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 25850* 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 25851 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 25852 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 25853 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 25854 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 25855 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 25856 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 25857 Lemma 1 for 1pthon 25859. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
𝐹 = {⟨0, 𝑖⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}       Fun (𝑃 ↾ (1..^(#‘𝐹)))

Theorem1pthonlem2 25858 Lemma 2 for 1pthon 25859. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
𝐹 = {⟨0, 𝑖⟩}    &   𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩}       ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅

Theorem1pthon 25859 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 25860 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 25861* 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 25862 Lemma 1 for constr2spth 25868. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → Fun 𝑃)

Theorem2pthlem1 25863 Lemma 1 for constr2pth 25869. (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 25864 Lemma 2 for constr2pth 25869. (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 25865* Lemma 1 for constr2wlk 25866. (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 25866 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 25867 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 25868 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 25869 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 25870 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 25871 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 25872* 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 25873 Lemma for redwlk 25874. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
((𝐹(𝑉 Walks 𝐸)𝑃 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1))

Theoremredwlk 25874 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 25875* Lemma for wlkdvspth 25876. (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 25876 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 25877 Lemma 1 for usgra2adedgspth 25879. (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 25878 Lemma 2 for usgra2adedgspth 25879. (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 25879 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 25880 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 25881 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 25882 Alternate proof for usgra2adedgwlkon 25881, using usgra2adedgwlk 25880, 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 25883* 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 25884* 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 25885* Lemma 1 for usgra2wlkspth 25887. (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 25886* Lemma 2 for usgra2wlkspth 25887. (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 25887 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))

16.1.5.3  Circuits and cycles

Theoremcrcts 25888* 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 25889* 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 25890 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 25891 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 25892 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 25893 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 25894 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝐹(𝑉 Circuits 𝐸)𝑃𝐹(𝑉 Trails 𝐸)𝑃)

Theoremcyclispth 25895 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Paths 𝐸)𝑃)

Theoremcycliscrct 25896 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Circuits 𝐸)𝑃)

Theoremcyclnspth 25897 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 25898 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.)
(𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremcyclispthon 25899 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 25900 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))

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