P. 298 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wlkelwrd 29701	The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉))
Theorem	wlkeq 29702*	Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))))
Theorem	edginwlk 29703	The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((Fun 𝐼 ∧ 𝐹 ∈ Word dom 𝐼 ∧ 𝐾 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸)
Theorem	upgredginwlk 29704	The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝐾 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸))
Theorem	iedginwlk 29705	The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 23-Apr-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((Fun 𝐼 ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑋)) ∈ ran 𝐼)
Theorem	wlkl1loop 29706	A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)
⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
Theorem	wlk1walk 29707*	A walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))
Theorem	wlk1ewlk 29708	A walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ (𝐺 EdgWalks 1))
Theorem	upgriswlk 29709*	Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
Theorem	upgrwlkedg 29710*	The edges of a walk in a pseudograph join exactly the two corresponding adjacent vertices in the walk. (Contributed by AV, 27-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
Theorem	upgrwlkcompim 29711*	Implications for the properties of the components of a walk in a pseudograph. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 14-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
Theorem	wlkvtxedg 29712*	The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ 𝐸 {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒)
Theorem	upgrwlkvtxedg 29713*	The pairs of connected vertices of a walk are edges in a pseudograph. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸)
Theorem	uspgr2wlkeq 29714*	Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.)
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))
Theorem	uspgr2wlkeq2 29715	Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝐵)) = 𝑁)) → ((2nd ‘𝐴) = (2nd ‘𝐵) → 𝐴 = 𝐵))
Theorem	uspgr2wlkeqi 29716	Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 6-May-2021.)
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) → 𝐴 = 𝐵)
Theorem	umgrwlknloop 29717*	In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 3-Jan-2021.)
⊢ ((𝐺 ∈ UMGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
Theorem	wlkv0 29718	If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st ‘𝑊) = ∅ ∧ (2nd ‘𝑊) = ∅))
Theorem	g0wlk0 29719	There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅)
Theorem	0wlk0 29720	There is no walk for the empty set, i.e. in a null graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
⊢ (Walks‘∅) = ∅
Theorem	wlk0prc 29721	There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
⊢ ((𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺)) → (Walks‘𝐺) = ∅)
Theorem	wlklenvclwlk 29722	The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.)
⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (♯‘𝐹) = (♯‘𝑊)))
Theorem	wlkson 29723*	The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)})
Theorem	iswlkon 29724	Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 31-Dec-2020.) (Revised by AV, 22-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))
Theorem	wlkonprop 29725	Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 31-Dec-2020.) (Proof shortened by AV, 16-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))
Theorem	wlkpvtx 29726	A walk connects vertices. (Contributed by AV, 22-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑁 ∈ (0...(♯‘𝐹)) → (𝑃‘𝑁) ∈ 𝑉))
Theorem	wlkepvtx 29727	The endpoints of a walk are vertices. (Contributed by AV, 31-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(♯‘𝐹)) ∈ 𝑉))
Theorem	wlkoniswlk 29728	A walk between two vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 2-Jan-2021.)
⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	wlkonwlk 29729	A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃)
Theorem	wlkonwlk1l 29730	A walk is a walk from its first vertex to its last vertex. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 22-Mar-2021.)
⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃)
Theorem	wlksoneq1eq2 29731	Two walks with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.)
⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐻(𝐶(WalksOn‘𝐺)𝐷)𝑃) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	wlkonl1iedg 29732*	If there is a walk between two vertices 𝐴 and 𝐵 at least of length 1, then the start vertex 𝐴 is incident with an edge. (Contributed by AV, 4-Apr-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ (♯‘𝐹) ≠ 0) → ∃𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒)
Theorem	wlkon2n0 29733	The length of a walk between two different vertices is not 0 (i.e. is at least 1). (Contributed by AV, 3-Apr-2021.)
⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐴 ≠ 𝐵) → (♯‘𝐹) ≠ 0)
Theorem	2wlklem 29734*	Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
Theorem	upgr2wlk 29735	Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
Theorem	wlkreslem 29736	Lemma for wlkres 29737. (Contributed by AV, 5-Mar-2021.) (Revised by AV, 30-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    ⇒   ⊢ (𝜑 → 𝑆 ∈ V)
Theorem	wlkres 29737	The restriction ⟨𝐻, 𝑄⟩ of a walk ⟨𝐹, 𝑃⟩ to an initial segment of the walk (of length 𝑁) forms a walk on the subgraph 𝑆 consisting of the edges in the initial segment. Formerly proven directly for Eulerian paths, see eupthres 30285. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ 𝐻 = (𝐹 prefix 𝑁)    &   ⊢ 𝑄 = (𝑃 ↾ (0...𝑁))    ⇒   ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)
Theorem	redwlklem 29738	Lemma for redwlk 29739. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
⊢ ((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (♯‘𝐹) ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → (𝑃 ↾ (0..^(♯‘𝐹))):(0...(♯‘(𝐹 ↾ (0..^((♯‘𝐹) − 1)))))⟶𝑉)
Theorem	redwlk 29739	A walk ending at the last but one vertex of the walk is a walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 1 ≤ (♯‘𝐹)) → (𝐹 ↾ (0..^((♯‘𝐹) − 1)))(Walks‘𝐺)(𝑃 ↾ (0..^(♯‘𝐹))))
Theorem	wlkp1lem1 29740	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    ⇒   ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
Theorem	wlkp1lem2 29741	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    ⇒   ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1))
Theorem	wlkp1lem3 29742	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    ⇒   ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))
Theorem	wlkp1lem4 29743	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    ⇒   ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
Theorem	wlkp1lem5 29744*	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘))
Theorem	wlkp1lem6 29745*	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘))))
Theorem	wlkp1lem7 29746	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    ⇒   ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁)))
Theorem	wlkp1lem8 29747*	Lemma for wlkp1 29748. (Contributed by AV, 6-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    &   ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))
Theorem	wlkp1 29748	Append one path segment (edge) 𝐸 from vertex (𝑃‘𝑁) to a vertex 𝐶 to a walk ⟨𝐹, 𝑃⟩ to become a walk ⟨𝐻, 𝑄⟩ of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. Formerly proven directly for Eulerian paths (for pseudographs), see eupthp1 30286. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Proof shortened by AV, 18-Apr-2021.) (Revised by AV, 8-Apr-2024.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))    &   ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    &   ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})    ⇒   ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)
Theorem	wlkdlem1 29749*	Lemma 1 for wlkd 29753. (Contributed by AV, 7-Feb-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ (𝜑 → 𝐹 ∈ Word V)    &   ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
Theorem	wlkdlem2 29750*	Lemma 2 for wlkd 29753. (Contributed by AV, 7-Feb-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ (𝜑 → 𝐹 ∈ Word V)    &   ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → (((♯‘𝐹) ∈ ℕ → (𝑃‘(♯‘𝐹)) ∈ (𝐼‘(𝐹‘((♯‘𝐹) − 1)))) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ∈ (𝐼‘(𝐹‘𝑘))))
Theorem	wlkdlem3 29751*	Lemma 3 for wlkd 29753. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ (𝜑 → 𝐹 ∈ Word V)    &   ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
Theorem	wlkdlem4 29752*	Lemma 4 for wlkd 29753. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 23-Jan-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ (𝜑 → 𝐹 ∈ Word V)    &   ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
Theorem	wlkd 29753*	Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ (𝜑 → 𝐹 ∈ Word V)    &   ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
17.3.2  Walks for loop-free graphs
Theorem	lfgrwlkprop 29754*	Two adjacent vertices in a walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
Theorem	lfgriswlk 29755*	Conditions for a pair of functions to be a walk in a loop-free graph. (Contributed by AV, 28-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
Theorem	lfgrwlknloop 29756*	In a loop-free graph, each walk has no loops! (Contributed by AV, 2-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
17.3.3  Trails
Syntax	ctrls 29757	Extend class notation with trails (within a graph).
class Trails
Syntax	ctrlson 29758	Extend class notation with trails between two vertices (within a graph).
class TrailsOn
Definition	df-trls 29759*	Define the set of all Trails (in an undirected graph). According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct. According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5. Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)
⊢ Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)})
Definition	df-trlson 29760*	Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)
⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)}))
Theorem	reltrls 29761	The set (Trails‘𝐺) of all trails on 𝐺 is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.)
⊢ Rel (Trails‘𝐺)
Theorem	trlsfval 29762*	The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)
⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)}
Theorem	istrl 29763	Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)
⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
Theorem	trliswlk 29764	A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.)
⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	trlf1 29765	The enumeration 𝐹 of a trail ⟨𝐹, 𝑃⟩ is injective. (Contributed by AV, 20-Feb-2021.) (Proof shortened by AV, 29-Oct-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
Theorem	trlreslem 29766	Lemma for trlres 29767. Formerly part of proof of eupthres 30285. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ 𝐻 = (𝐹 prefix 𝑁)    ⇒   ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
Theorem	trlres 29767	The restriction ⟨𝐻, 𝑄⟩ of a trail ⟨𝐹, 𝑃⟩ to an initial segment of the trail (of length 𝑁) forms a trail on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ 𝐻 = (𝐹 prefix 𝑁)    &   ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ 𝑄 = (𝑃 ↾ (0...𝑁))    ⇒   ⊢ (𝜑 → 𝐻(Trails‘𝑆)𝑄)
Theorem	upgrtrls 29768*	The set of trails in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ ((𝑓 ∈ Word dom 𝐼 ∧ Fun ◡𝑓) ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
Theorem	upgristrl 29769*	Properties of a pair of functions to be a trail in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 29-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹) ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
Theorem	upgrf1istrl 29770*	Properties of a pair of a one-to-one function into the set of indices of edges and a function into the set of vertices to be a trail in a pseudograph. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 29-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
Theorem	wksonproplem 29771*	Lemma for theorems for properties of walks between two vertices, e.g., trlsonprop 29774. (Contributed by AV, 16-Jan-2021.) Remove is-walk hypothesis. (Revised by SN, 13-Dec-2024.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)))    &   ⊢ 𝑊 = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ∧ 𝑓(𝑄‘𝑔)𝑝)}))    ⇒   ⊢ (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)))
Theorem	trlsonfval 29772*	The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 15-Jan-2021.) (Revised by AV, 21-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Trails‘𝐺)𝑝)})
Theorem	istrlson 29773	Properties of a pair of functions to be a trail between two given vertices. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 21-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
Theorem	trlsonprop 29774	Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 16-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Trails‘𝐺)𝑃)))
Theorem	trlsonistrl 29775	A trail between two vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 7-Jan-2021.)
⊢ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → 𝐹(Trails‘𝐺)𝑃)
Theorem	trlsonwlkon 29776	A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.)
⊢ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃)
Theorem	trlontrl 29777	A trail is a trail between its endpoints. (Contributed by AV, 31-Jan-2021.)
⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹((𝑃‘0)(TrailsOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃)
17.3.4  Paths and simple paths
Syntax	cpths 29778	Extend class notation with paths (of a graph).
class Paths
Syntax	cspths 29779	Extend class notation with simple paths (of a graph).
class SPaths
Syntax	cpthson 29780	Extend class notation with paths between two vertices (within a graph).
class PathsOn
Syntax	cspthson 29781	Extend class notation with simple paths between two vertices (within a graph).
class SPathsOn
Definition	df-pths 29782*	Define the set of all paths (in an undirected graph). According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices." According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see upgrwlkdvspth 29807). Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)
⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
Definition	df-spths 29783*	Define the set of all simple paths (in an undirected graph). According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices." Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.)
⊢ SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)})
Definition	df-pthson 29784*	Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)
⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Paths‘𝑔)𝑝)}))
Definition	df-spthson 29785*	Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 9-Jan-2021.)
⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPaths‘𝑔)𝑝)}))
Theorem	relpths 29786	The set (Paths‘𝐺) of all paths on 𝐺 is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021.)
⊢ Rel (Paths‘𝐺)
Theorem	pthsfval 29787*	The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
⊢ (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}
Theorem	spthsfval 29788*	The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)}
Theorem	ispth 29789	Conditions for a pair of classes/functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅))
Theorem	isspth 29790	Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃))
Theorem	pthistrl 29791	A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
Theorem	spthispth 29792	A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃)
Theorem	pthiswlk 29793	A path is a walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)
⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	spthiswlk 29794	A simple path is a walk (in an undirected graph). (Contributed by AV, 16-May-2021.)
⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	pthdivtx 29795	The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))
Theorem	pthdadjvtx 29796	The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)
⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 𝐼 ∈ (0..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))
Theorem	dfpth2 29797	Alternate definition for a pair of classes/functions to be a path (in an undirected graph). (Contributed by AV, 4-Oct-2025.)
⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
Theorem	pthdifv 29798	The vertices of a path are distinct (except the first and last vertex), so the restricted vertex function is one-to-one. (Contributed by AV, 2-Oct-2025.)
⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺))
Theorem	2pthnloop 29799*	A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 30221. (Contributed by AV, 6-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))))
Theorem	upgr2pthnlp 29800*	A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))(♯‘(𝐼‘(𝐹‘𝑖))) = 2)