P. 304 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)

Type	Label	Description
Statement
17.3.7  Walks as words In general, a walk is an alternating sequence of vertices and edges, as defined in df-wlks 30067: 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 14583, 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 30306 and df-wwlksn 30307, 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).
Syntax	cwwlks 30301	Extend class notation with walks (in a graph) as word over the set of vertices.
class WWalks
Syntax	cwwlksn 30302	Extend class notation with walks (in a graph) of a fixed length as word over the set of vertices.
class WWalksN
Syntax	cwwlksnon 30303	Extend class notation with walks between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WWalksNOn
Syntax	cwwspthsn 30304	Extend class notation with simple paths (in a graph) of a fixed length as word over the set of vertices.
class WSPathsN
Syntax	cwwspthsnon 30305	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
Definition	df-wwlks 30306*	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-wlks 30067. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 30088. (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)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
Definition	df-wwlksn 30307*	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-wlks 30067. (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)})
Definition	df-wwlksnon 30308*	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 ∧ (𝑤‘𝑛) = 𝑏)}))
Definition	df-wspthsn 30309*	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})
Definition	df-wspthsnon 30310*	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}))
Theorem	wwlks 30311*	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)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}
Theorem	iswwlks 30312*	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)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
Theorem	wwlksn 30313*	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)})
Theorem	iswwlksn 30314	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))))
Theorem	wwlksnprcl 30315	Derivation of the length of a word 𝑊 whose concatenation with a singleton word represents a walk of a fixed length 𝑁 (a partial reverse closure theorem). (Contributed by AV, 4-Mar-2022.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → (♯‘𝑊) = 𝑁))
Theorem	iswwlksnx 30316*	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)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1))))
Theorem	wwlkbp 30317	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))
Theorem	wwlknbp 30318	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 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))
Theorem	wwlknp 30319*	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){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
Theorem	wwlknbp1 30320	Other basic properties of a walk of a fixed length as word. (Contributed by AV, 8-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))
Theorem	wwlknvtx 30321*	The symbols of a word 𝑊 representing a walk of a fixed length 𝑁 are vertices. (Contributed by AV, 16-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑖 ∈ (0...𝑁)(𝑊‘𝑖) ∈ (Vtx‘𝐺))
Theorem	wwlknllvtx 30322	If a word 𝑊 represents a walk of a fixed length 𝑁, then the first and the last symbol of the word is a vertex. (Contributed by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) ∈ 𝑉 ∧ (𝑊‘𝑁) ∈ 𝑉))
Theorem	wwlknlsw 30323	If a word represents a walk of a fixed length, then the last symbol of the word is the endvertex of the walk. (Contributed by AV, 8-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊‘𝑁) = (lastS‘𝑊))
Theorem	wspthsn 30324*	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}
Theorem	iswspthn 30325*	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))
Theorem	wspthnp 30326*	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 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
Theorem	wwlksnon 30327*	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 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}))
Theorem	wspthsnon 30328*	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 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
Theorem	iswwlksnon 30329*	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.) (Revised by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}
Theorem	wwlksnon0 30330	Sufficient conditions for a set of walks of a fixed length between two vertices to be empty. (Contributed by AV, 15-May-2021.) (Proof shortened by AV, 21-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (¬ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
Theorem	wwlksonvtx 30331	If a word 𝑊 represents a walk of length 2 on two classes 𝐴 and 𝐶, these classes are vertices. (Contributed by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
Theorem	iswspthsnon 30332*	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.) (Revised by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
Theorem	wwlknon 30333	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.) (Revised by AV, 14-Mar-2022.)
⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘𝑁) = 𝐵))
Theorem	wspthnon 30334*	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.) (Revised by AV, 15-Mar-2022.)
⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
Theorem	wspthnonp 30335*	Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021.) (Proof shortened by AV, 15-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))
Theorem	wspthneq1eq2 30336	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 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	wwlksn0s 30337*	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}
Theorem	wwlkssswrd 30338	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
Theorem	wwlksn0 30339*	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) → ∃𝑣 ∈ 𝑉 𝑊 = ⟨“𝑣”⟩)
Theorem	0enwwlksnge1 30340	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) = ∅)
Theorem	wwlkswwlksn 30341	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))
Theorem	wwlkssswwlksn 30342	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)
Theorem	wlkiswwlks1 30343	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(Walks‘𝐺)𝑃 → 𝑃 ∈ (WWalks‘𝐺)))
Theorem	wlklnwwlkln1 30344	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(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 𝑁) → 𝑃 ∈ (𝑁 WWalksN 𝐺)))
Theorem	wlkiswwlks2lem1 30345*	Lemma 1 for wlkiswwlks2 30351. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1))
Theorem	wlkiswwlks2lem2 30346*	Lemma 2 for wlkiswwlks2 30351. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    ⇒   ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}))
Theorem	wlkiswwlks2lem3 30347*	Lemma 3 for wlkiswwlks2 30351. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
Theorem	wlkiswwlks2lem4 30348*	Lemma 4 for wlkiswwlks2 30351. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
Theorem	wlkiswwlks2lem5 30349*	Lemma 5 for wlkiswwlks2 30351. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → 𝐹 ∈ Word dom 𝐸))
Theorem	wlkiswwlks2lem6 30350*	Lemma 6 for wlkiswwlks2 30351. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
Theorem	wlkiswwlks2 30351*	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(Walks‘𝐺)𝑃))
Theorem	wlkiswwlks 30352*	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(Walks‘𝐺)𝑃 ↔ 𝑃 ∈ (WWalks‘𝐺)))
Theorem	wlkiswwlksupgr2 30353*	A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 30351 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 10486) for its proof. Notice that only the existence of a function 𝑓 can be proven, but, in general, it cannot be "constructed" (as in wlkiswwlks2 30351). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
Theorem	wlkiswwlkupgr 30354*	A walk as word corresponds to a walk in a pseudograph. This variant of wlkiswwlks 30352 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(Walks‘𝐺)𝑃 ↔ 𝑃 ∈ (WWalks‘𝐺)))
Theorem	wlkswwlksf1o 30355*	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 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑤))    ⇒   ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
Theorem	wlkswwlksen 30356	The set of walks as words and the set of (ordinary) walks are equinumerous in a simple pseudograph. (Contributed by AV, 6-May-2021.) (Revised by AV, 5-Aug-2022.)
⊢ (𝐺 ∈ USPGraph → (Walks‘𝐺) ≈ (WWalks‘𝐺))
Theorem	wwlksm1edg 30357	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.) (Revised by AV, 26-Oct-2022.)
⊢ ((𝑊 ∈ (WWalks‘𝐺) ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix ((♯‘𝑊) − 1)) ∈ (WWalks‘𝐺))
Theorem	wlklnwwlkln2lem 30358*	Lemma for wlklnwwlkln2 30359 and wlklnwwlklnupgr2 30361. Formerly part of proof for wlklnwwlkln2 30359. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
⊢ (𝜑 → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))    ⇒   ⊢ (𝜑 → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
Theorem	wlklnwwlkln2 30359*	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(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
Theorem	wlklnwwlkn 30360*	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(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
Theorem	wlklnwwlklnupgr2 30361*	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 wlklnwwlkln2 30359 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(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
Theorem	wlklnwwlknupgr 30362*	A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a pseudograph. This variant of wlklnwwlkn 30360 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(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
Theorem	wlknewwlksn 30363	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 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st ‘𝑊)) = 𝑁)) → (2nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺))
Theorem	wlknwwlksnbij 30364*	The mapping (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) 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, 5-Aug-2022.)
⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}    &   ⊢ 𝑊 = (𝑁 WWalksN 𝐺)    &   ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)
Theorem	wlknwwlksnen 30365*	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, 5-Aug-2022.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))
Theorem	wlknwwlksneqs 30366*	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 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = (♯‘(𝑁 WWalksN 𝐺)))
Theorem	wwlkseq 30367*	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))(𝑊‘𝑖) = (𝑇‘𝑖))))
Theorem	wwlksnred 30368	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.) (Revised by AV, 26-Oct-2022.)
⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
Theorem	wwlksnext 30369	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) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
Theorem	wwlksnextbi 30370	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.) (Proof shortened by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
Theorem	wwlksnredwwlkn 30371*	For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
Theorem	wwlksnredwwlkn0 30372*	For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
Theorem	wwlksnextwrd 30373*	Lemma for wwlksnextbij 30378. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
Theorem	wwlksnextfun 30374*	Lemma for wwlksnextbij 30378. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)
Theorem	wwlksnextinj 30375*	Lemma for wwlksnextbij 30378. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷–1-1→𝑅)
Theorem	wwlksnextsurj 30376*	Lemma for wwlksnextbij 30378. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)
Theorem	wwlksnextbij0 30377*	Lemma for wwlksnextbij 30378. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅)
Theorem	wwlksnextbij 30378*	There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
Theorem	wwlksnexthasheq 30379*	The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}) = (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))
Theorem	disjxwwlksn 30380*	Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 29-Jul-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}
Theorem	wwlksnndef 30381	Conditions for WWalksN not being defined. (Contributed by Alexander van der Vekens, 30-Jul-2018.) (Revised by AV, 19-Apr-2021.)
⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ0) → (𝑁 WWalksN 𝐺) = ∅)
Theorem	wwlksnfi 30382	The number of walks represented by words of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 30-Jul-2018.) (Revised by AV, 19-Apr-2021.) (Proof shortened by JJ, 18-Nov-2022.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
Theorem	wlksnfi 30383*	The number of walks of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 20-Apr-2021.)
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ∈ Fin)
Theorem	wlksnwwlknvbij 30384*	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 and starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 5-Aug-2022.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
Theorem	wwlksnextproplem1 30385	Lemma 1 for wwlksnextprop 30388. (Contributed by Alexander van der Vekens, 31-Jul-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))
Theorem	wwlksnextproplem2 30386	Lemma 2 for wwlksnextprop 30388. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)
Theorem	wwlksnextproplem3 30387*	Lemma 3 for wwlksnextprop 30388. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ0) → (𝑊 prefix (𝑁 + 1)) ∈ 𝑌)
Theorem	wwlksnextprop 30388*	Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
Theorem	disjxwwlkn 30389*	Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}
Theorem	hashwwlksnext 30390*	Number of walks (as words) extended by an edge as a sum over the prefixes. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}))
Theorem	wwlksnwwlksnon 30391*	A walk of fixed length is a walk of fixed length between two vertices. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 13-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏))
Theorem	wspthsnwspthsnon 30392*	A simple path of fixed length is a simple path of fixed length between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-May-2021.) (Revised by AV, 13-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WSPathsNOn 𝐺)𝑏))
Theorem	wspthsnonn0vne 30393	If the set of simple paths of length at least 1 between two vertices is not empty, the two vertices must be different. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑋(𝑁 WSPathsNOn 𝐺)𝑌) ≠ ∅) → 𝑋 ≠ 𝑌)
Theorem	wspthsswwlkn 30394	The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021.)
⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)
Theorem	wspthnfi 30395	In a finite graph, the set of simple paths of a fixed length is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 18-May-2021.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WSPathsN 𝐺) ∈ Fin)
Theorem	wwlksnonfi 30396	In a finite graph, the set of walks of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 15-May-2021.) (Proof shortened by AV, 15-Mar-2022.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ Fin)
Theorem	wspthsswwlknon 30397	The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length between the two vertices. (Contributed by AV, 15-May-2021.)
⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ⊆ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)
Theorem	wspthnonfi 30398	In a finite graph, the set of simple paths of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 15-May-2021.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∈ Fin)
Theorem	wspniunwspnon 30399*	The set of nonempty simple paths of fixed length is the double union of the simple paths of the fixed length between different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 15-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦))
Theorem	wspn0 30400	If there are no vertices, then there are no simple paths (of any length), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 13-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑉 = ∅ → (𝑁 WSPathsN 𝐺) = ∅)