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Theorem List for Metamath Proof Explorer - 41001-41100   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremcrctisTrl 41001 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(CircuitS‘𝐺)𝑃𝐹(TrailS‘𝐺)𝑃)

Theoremcrctis1wlk 41002 A circuit is a walk. (Contributed by AV, 6-Apr-2021.)
(𝐹(CircuitS‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)

TheoremcyclisPth 41003 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(CycleS‘𝐺)𝑃𝐹(PathS‘𝐺)𝑃)

TheoremcyclisWlk 41004 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(CycleS‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)

TheoremcyclisCrct 41005 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(CycleS‘𝐺)𝑃𝐹(CircuitS‘𝐺)𝑃)

TheoremcyclnsPth 41006 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹 ≠ ∅ → (𝐹(CycleS‘𝐺)𝑃 → ¬ 𝐹(SPathS‘𝐺)𝑃))

TheoremcyclisPthon 41007 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(CycleS‘𝐺)𝑃𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘0))𝑃)

Theoremlfgrn1cycl 41008* In a loop-free graph there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → (𝐹(CycleS‘𝐺)𝑃 → (#‘𝐹) ≠ 1))

Theoremusgr2trlncrct 41009 In a simple graph, any trail of length 2 is not a circuit. (Contributed by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(TrailS‘𝐺)𝑃 → ¬ 𝐹(CircuitS‘𝐺)𝑃))

Theoremumgrn1cycl 41010 In a multigraph graph (with no loops!) there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
((𝐺 ∈ UMGraph ∧ 𝐹(CycleS‘𝐺)𝑃) → (#‘𝐹) ≠ 1)

Theoremuspgrn2crct 41011 In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.)
((𝐺 ∈ USPGraph ∧ 𝐹(CircuitS‘𝐺)𝑃) → (#‘𝐹) ≠ 2)

Theoremusgrn2cycl 41012 In a simple graph there are no cycles with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 4-Feb-2021.)
((𝐺 ∈ USGraph ∧ 𝐹(CycleS‘𝐺)𝑃) → (#‘𝐹) ≠ 2)

Theoremcrctcsh1wlkn0lem1 41013 Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 13-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ) → ((𝐴𝐵) + 1) ≤ 𝐴)

Theoremcrctcsh1wlkn0lem2 41014* Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝐽 ∈ (0...(𝑁𝑆))) → (𝑄𝐽) = (𝑃‘(𝐽 + 𝑆)))

Theoremcrctcsh1wlkn0lem3 41015* Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝐽 ∈ (((𝑁𝑆) + 1)...𝑁)) → (𝑄𝐽) = (𝑃‘((𝐽 + 𝑆) − 𝑁)))

Theoremcrctcsh1wlkn0lem4 41016* Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))       (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))

Theoremcrctcsh1wlkn0lem5 41017* Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))       (𝜑 → ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))

Theoremcrctcsh1wlkn0lem6 41018* Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))    &   (𝜑 → (𝑃𝑁) = (𝑃‘0))       ((𝜑𝐽 = (𝑁𝑆)) → if-((𝑄𝐽) = (𝑄‘(𝐽 + 1)), (𝐼‘(𝐻𝐽)) = {(𝑄𝐽)}, {(𝑄𝐽), (𝑄‘(𝐽 + 1))} ⊆ (𝐼‘(𝐻𝐽))))

Theoremcrctcsh1wlkn0lem7 41019* Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))    &   (𝜑 → (𝑃𝑁) = (𝑃‘0))       (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))

Theoremcrctcshlem1 41020 Lemma for crctcsh 41027. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)       (𝜑𝑁 ∈ ℕ0)

Theoremcrctcshlem2 41021 Lemma for crctcsh 41027. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)       (𝜑 → (#‘𝐻) = 𝑁)

Theoremcrctcshlem3 41022* Lemma for crctcsh 41027. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))

Theoremcrctcshlem4 41023* Lemma for crctcsh 41027. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝑆 = 0) → (𝐻 = 𝐹𝑄 = 𝑃))

Theoremcrctcsh1wlkn0 41024* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a 1-walk 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝑆 ≠ 0) → 𝐻(1Walks‘𝐺)𝑄)

Theoremcrctcsh1wlk 41025* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a 1-walk 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(1Walks‘𝐺)𝑄)

Theoremcrctcshtrl 41026* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a trail 𝐻, 𝑄. (Contributed by AV, 14-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(TrailS‘𝐺)𝑄)

Theoremcrctcsh 41027* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a circuit 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(CircuitS‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(CircuitS‘𝐺)𝑄)

21.34.8.15  Walks as words

In general, a walk is an alternating sequence of vertices and edges, as defined in df-1wlks 40800: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Often, it is sufficient to refer to a walk by the natural sequence of its vertices, i.e omitting its edges in its representation: p(0) p(1) ... p(n-1) p(n), see the corresponding remark in [Diestel] p. 6. The concept of a Word, see df-word 13154, is the appropriate way to define such a sequence (being finite and starting at index 0) of vertices. Therefore, it is used in definitions df-wwlks 41033 and df-wwlksn 41034, and the representation of a walk as sequence of its vertices is called "walk as word".

Only for simple pseudographs, however, the edges can be uniquely reconstructed from such a representation. In other cases, there could be more than one edge between two adjacent vertices in the walk (in a multigraph), or two adjacent vertices could be connected by two different hyperedges involving additional vertices (in a hypergraph).

Syntaxcwwlks 41028 Extend class notation with walks (in a graph) as word over the set of vertices.
class WWalkS

Syntaxcwwlksn 41029 Extend class notation with walks (in a graph) of a fixed length as word over the set of vertices.
class WWalkSN

Syntaxcwwlksnon 41030 Extend class notation with walks between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WWalksNOn

Syntaxcwwspthsn 41031 Extend class notation with simple paths (in a graph) of a fixed length as word over the set of vertices.
class WSPathsN

Syntaxcwwspthsnon 41032 Extend class notation with simple paths between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WSPathsNOn

Definitiondf-wwlks 41033* Define the set of all walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-1wlks 40800. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 26055. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalkS = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})

Definitiondf-wwlksn 41034* Define the set of all walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-1wlks 40800. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalkSN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})

Definitiondf-wwlksnon 41035* Define the collection of walks of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))

Definitiondf-wspthsn 41036* Define the collection of simple paths of a fixed length as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ∃𝑓 𝑓(SPathS‘𝑔)𝑤})

Definitiondf-wspthsnon 41037* Define the collection of simple paths of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))

Theoremwwlks 41038* The set of walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (WWalkS‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}

Theoremiswwlks 41039* A word over the set of vertices representing a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (WWalkS‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))

Theoremwwlksn 41040* The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})

Theoremiswwlksn 41041 A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) ↔ (𝑊 ∈ (WWalkS‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))))

Theoremiswwlksnx 41042* Properties of a word to represent a walk of a fixed length, definition of WWalkS expanded. (Contributed by AV, 28-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (#‘𝑊) = (𝑁 + 1))))

Theoremwwlkbp 41043 Basic properties of a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (WWalkS‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉))

Theoremwwlknbp 41044 Basic properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 20-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))

Theoremwwlknp 41045* Properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))

Theoremwspthsn 41046* The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ∃𝑓 𝑓(SPathS‘𝐺)𝑤}

Theoremiswspthn 41047* An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊))

Theoremwspthnp 41048* Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ ∃𝑓 𝑓(SPathS‘𝐺)𝑊))

Theoremwwlksnon 41049* The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))

Theoremwspthsnon 41050* The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))

Theoremiswwlksnon 41051* The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})

Theoremiswspthsnon 41052* The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})

Theoremwwlknon 41053 An element of the set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))

Theoremwspthnon 41054* An element of the set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))

Theoremwspthnonp 41055* Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))

Theoremwspthneq1eq2 41056 Two simple paths with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.)
((𝑃 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ 𝑃 ∈ (𝐶(𝑁 WSPathsNOn 𝐺)𝐷)) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremwwlksn0s 41057* The set of all walks as words of length 0 is the set of all words of length 1 over the vertices. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(0 WWalkSN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (#‘𝑤) = 1}

Theoremwwlkssswrd 41058 Walks (represented by words) are words. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (WWalkS‘𝐺) ⊆ Word 𝑉

Theoremwwlksn0 41059* A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 21-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (0 WWalkSN 𝐺) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)

Theorem0enwwlksnge1 41060 In graphs without edges, there are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 7-May-2021.)
(((Edg‘𝐺) = ∅ ∧ 𝑁 ∈ ℕ) → (𝑁 WWalkSN 𝐺) = ∅)

Theoremwwlkswwlksn 41061 A walk of a fixed length as word is a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝑊 ∈ (WWalkS‘𝐺))

Theoremwwlkssswwlksn 41062 The walks of a fixed length as words are walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝑁 WWalkSN 𝐺) ⊆ (WWalkS‘𝐺)

Theoremwwlknbp2 41063 Other basic properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 12-Apr-2021.)
(𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))

Theorem1wlkiswwlks1 41064 The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)
(𝐺 ∈ UPGraph → (𝐹(1Walks‘𝐺)𝑃𝑃 ∈ (WWalkS‘𝐺)))

Theorem1wlklnwwlkln1 41065 The sequence of vertices in a walk of length 𝑁 is a walk as word of length 𝑁 in a pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → ((𝐹(1Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 𝑁) → 𝑃 ∈ (𝑁 WWalkSN 𝐺)))

Theorem1wlkiswwlks2lem1 41066* Lemma 1 for 1wlkiswwlks2 41072. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘𝐹) = ((#‘𝑃) − 1))

Theorem1wlkiswwlks2lem2 41067* Lemma 2 for 1wlkiswwlks2 41072. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       (((#‘𝑃) ∈ ℕ0𝐼 ∈ (0..^((#‘𝑃) − 1))) → (𝐹𝐼) = (𝐸‘{(𝑃𝐼), (𝑃‘(𝐼 + 1))}))

Theorem1wlkiswwlks2lem3 41068* Lemma 3 for 1wlkiswwlks2 41072. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃:(0...(#‘𝐹))⟶𝑉)

Theorem1wlkiswwlks2lem4 41069* Lemma 4 for 1wlkiswwlks2 41072. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))

Theorem1wlkiswwlks2lem5 41070* Lemma 5 for 1wlkiswwlks2 41072. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸𝐹 ∈ Word dom 𝐸))

Theorem1wlkiswwlks2lem6 41071* Lemma 6 for 1wlkiswwlks2 41072. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))

Theorem1wlkiswwlks2 41072* A walk as word corresponds to the sequence of vertices in a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ USPGraph → (𝑃 ∈ (WWalkS‘𝐺) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃))

Theorem1wlkiswwlks 41073* A walk as word corresponds to a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ USPGraph → (∃𝑓 𝑓(1Walks‘𝐺)𝑃𝑃 ∈ (WWalkS‘𝐺)))

Theorem1wlkiswwlksupgr2 41074* A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of 1wlkiswwlks2 41072 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 9185) for its proof. Notice that only the existence of a function 𝑓 can be proven, but, in general, it cannot be "constructed" (as in 1wlkiswwlks2 41072). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ UPGraph → (𝑃 ∈ (WWalkS‘𝐺) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃))

Theorem1wlkiswwlkupgr 41075* A walk as word corresponds to a walk in a pseudograph. This variant of 1wlkiswwlks 41073 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ UPGraph → (∃𝑓 𝑓(1Walks‘𝐺)𝑃𝑃 ∈ (WWalkS‘𝐺)))

Theorem1wlkpwwlkf1ouspgr 41076* The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)
𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd𝑤))       (𝐺 ∈ USPGraph → 𝐹:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺))

Theorem1wlkisowwlkupgr 41077* The set of walks as words and the set of (ordinary) walks are isomorphic in a simple pseudograph. (Contributed by AV, 6-May-2021.)
(𝐺 ∈ USPGraph → ∃𝑓 𝑓:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺))

Theoremwwlksm1edg 41078 Removing the trailing edge from a walk (as word) with at least one edge results in a walk. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 19-Apr-2021.)
((𝑊 ∈ (WWalkS‘𝐺) ∧ 2 ≤ (#‘𝑊)) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ∈ (WWalkS‘𝐺))

Theorem1wlklnwwlkln2lem 41079* Lemma for 1wlklnwwlkln2 41080 and 1wlklnwwlklnupgr2 41082. Formerly part of proof for 1wlklnwwlkln2 41080. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝜑 → (𝑃 ∈ (WWalkS‘𝐺) → ∃𝑓 𝑓(1Walks‘𝐺)𝑃))       (𝜑 → (𝑃 ∈ (𝑁 WWalkSN 𝐺) → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))

Theorem1wlklnwwlkln2 41080* A walk of length 𝑁 as word corresponds to the sequence of vertices in a walk of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ USPGraph → (𝑃 ∈ (𝑁 WWalkSN 𝐺) → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))

Theorem1wlklnwwlkn 41081* A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ USPGraph → (∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalkSN 𝐺)))

Theorem1wlklnwwlklnupgr2 41082* A walk of length 𝑁 as word corresponds to the sequence of vertices in a walk of length 𝑁 in a pseudograph. This variant of 1wlklnwwlkln2 41080 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → (𝑃 ∈ (𝑁 WWalkSN 𝐺) → ∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁)))

Theorem1wlklnwwlknupgr 41083* A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a pseudograph. This variant of 1wlkiswwlks 40197 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → (∃𝑓(𝑓(1Walks‘𝐺)𝑃 ∧ (#‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalkSN 𝐺)))

Theoremwlknewwlksn 41084 If a walk in a pseudograph has length 𝑁, then the sequence of the vertices of the walk is a word representing the walk as word of length 𝑁. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 11-Apr-2021.)
(((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ (𝑁 WWalkSN 𝐺))

Theoremwlknwwlksnfun 41085* Lemma 1 for wlknwwlksnbij2 41089. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalkSN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)

Theoremwlknwwlksninj 41086* Lemma 2 for wlknwwlksnbij2 41089. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalkSN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlknwwlksnsur 41087* Lemma 3 for wlknwwlksnbij2 41089. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalkSN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlknwwlksnbij 41088* Lemma 4 for wlknwwlksnbij2 41089. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalkSN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlknwwlksnbij2 41089* There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}–1-1-onto→(𝑁 WWalkSN 𝐺))

Theoremwlknwwlksnen 41090* In a simple pseudograph, the set of walks of a fixed length and the set of walks represented by words are equinumerous. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁} ≈ (𝑁 WWalkSN 𝐺))

Theoremwlknwwlksneqs 41091* The set of walks of a fixed length and the set of walks represented by words have the same size. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (#‘{𝑝 ∈ (1Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}) = (#‘(𝑁 WWalkSN 𝐺)))

Theoremwlkwwlkfun 41092* Lemma 1 for wlkwwlkbij2 41096. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)

Theoremwlkwwlkinj 41093* Lemma 2 for wlkwwlkbij2 41096. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlkwwlksur 41094* Lemma 3 for wlkwwlkbij2 41096. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Revised by AV, 16-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlkwwlkbij 41095* Lemma 4 for wlkwwlkbij2 41096. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.)
𝑇 = {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlkwwlkbij2 41096* There is a bijection between the set of walks of a fixed length, starting at a fixed vertex, and the set of walks represented as words of the same length, starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃})

Theoremwwlkseq 41097* Equality of two walks (as words). (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
((𝑊 ∈ (WWalkS‘𝐺) ∧ 𝑇 ∈ (WWalkS‘𝐺)) → (𝑊 = 𝑇 ↔ ((#‘𝑊) = (#‘𝑇) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑇𝑖))))

Theoremwwlksnred 41098 Reduction of a walk (as word) by removing the trailing edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalkSN 𝐺)))

Theoremwwlksnext 41099 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑇 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺))

Theoremwwlksnextbi 41100 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalkSN 𝐺)))

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