P. 302 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30101-30200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wlkeq 30101*	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 30102	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 30103	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 30104	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 30105	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 30106*	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 30107	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 30108*	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 30109*	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 30110*	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 30111*	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 30112*	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 30113*	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 30114	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 30115	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 30116*	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 30117	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 30118	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 30119	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 30120	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 30121	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 30122*	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 30123	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 30124	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 30125	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 30126	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 30127	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 30128	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 30129	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 30130	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 30131*	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 30132	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 30133*	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 30134	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 30135	Lemma for wlkres 30136. (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 30136	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 30703. (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 30137	Lemma for redwlk 30138. (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 30138	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 30139	Lemma for wlkp1 30147. (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 30140	Lemma for wlkp1 30147. (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 30141	Lemma for wlkp1 30147. (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 30142	Lemma for wlkp1 30147. (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 30143*	Lemma for wlkp1 30147. (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 30144*	Lemma for wlkp1 30147. (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 30145	Lemma for wlkp1 30147. (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 30146*	Lemma for wlkp1 30147. (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 30147	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 30704. (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 30148*	Lemma 1 for wlkd 30152. (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 30149*	Lemma 2 for wlkd 30152. (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 30150*	Lemma 3 for wlkd 30152. (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 30151*	Lemma 4 for wlkd 30152. (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 30152*	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)
Theorem	pfxwlk 30153	A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))
Theorem	revwlk 30154	The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.)
⊢ (𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃))
Theorem	swrdwlk 30155	Two matching subwords of a walk also represent a walk. (Contributed by BTernaryTau, 7-Dec-2023.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐵 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 substr ⟨𝐵, 𝐿⟩)(Walks‘𝐺)(𝑃 substr ⟨𝐵, (𝐿 + 1)⟩))
Theorem	subgrwlk 30156	If a walk exists in a subgraph of a graph 𝐺, then that walk also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
⊢ (𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃 → 𝐹(Walks‘𝐺)𝑃))
17.3.2  Walks for loop-free graphs
Theorem	lfgrwlkprop 30157*	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 30158*	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 30159*	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 30160	Extend class notation with trails (within a graph).
class Trails
Syntax	ctrlson 30161	Extend class notation with trails between two vertices (within a graph).
class TrailsOn
Definition	df-trls 30162*	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 30163*	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 30164	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 30165*	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 30166	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 30167	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 30168	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 30169	Lemma for trlres 30170. Formerly part of proof of eupthres 30703. (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 30170	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 30171*	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 30172*	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 30173*	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 30174*	Lemma for theorems for properties of walks between two vertices, e.g., trlsonprop 30177. (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 30175*	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 30176	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 30177	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 30178	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 30179	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 30180	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)
Theorem	subgrtrl 30181	If a trail exists in a subgraph of a graph 𝐺, then that trail also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
⊢ (𝑆 SubGraph 𝐺 → (𝐹(Trails‘𝑆)𝑃 → 𝐹(Trails‘𝐺)𝑃))
17.3.4  Paths and simple paths
Syntax	cpths 30182	Extend class notation with paths (of a graph).
class Paths
Syntax	cspths 30183	Extend class notation with simple paths (of a graph).
class SPaths
Syntax	cpthson 30184	Extend class notation with paths between two vertices (within a graph).
class PathsOn
Syntax	cspthson 30185	Extend class notation with simple paths between two vertices (within a graph).
class SPathsOn
Definition	df-pths 30186*	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 30212). 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 30187*	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 30188*	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 30189*	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 30190	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 30191*	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 30192*	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 30193	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 30194	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 30195	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 30196	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 30197	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 30198	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 30199	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 30200	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)))