P. 165 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16401-16500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	umgrspan 16401	A spanning subgraph S of a multigraph G is a multigraph. (Contributed by AV, 27-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   &    |- ( ph -> S e. W )   &    |- ( ph -> (Vtx ` S ) = V )   &    |- ( ph -> (iEdg ` S ) = ( E |` A ) )   &    |- ( ph -> G e. UMGraph )   =>    |- ( ph -> S e. UMGraph )
Theorem	usgrspan 16402	A spanning subgraph S of a simple graph G is a simple graph. (Contributed by AV, 15-Oct-2020.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   &    |- ( ph -> S e. W )   &    |- ( ph -> (Vtx ` S ) = V )   &    |- ( ph -> (iEdg ` S ) = ( E |` A ) )   &    |- ( ph -> G e. USGraph )   =>    |- ( ph -> S e. USGraph )
Theorem	uhgrspanop 16403	A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. UHGraph -> <. V , ( E |` A ) >. e. UHGraph )
Theorem	upgrspanop 16404	A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. UPGraph -> <. V , ( E |` A ) >. e. UPGraph )
Theorem	umgrspanop 16405	A spanning subgraph of a multigraph represented by an ordered pair is a multigraph. (Contributed by AV, 27-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. UMGraph -> <. V , ( E |` A ) >. e. UMGraph )
Theorem	usgrspanop 16406	A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   =>    |- ( G e. USGraph -> <. V , ( E |` A ) >. e. USGraph )
12.2.8  Vertex degree
Syntax	cvtxdg 16407	Extend class notation with the vertex degree function.
class VtxDeg
Definition	df-vtxdg 16408*	Define the vertex degree function for a graph. To be appropriate for arbitrary hypergraphs, we have to double-count those edges that contain u "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is infinite), the extended addition +e is used for the summation of the number of "ordinary" edges" and the number of "loops". Because we cannot in general show that an arbitrary set is either finite or infinite (see inffiexmid 7179), this definition is not as general as it may appear. But we keep it for consistency with the Metamath Proof Explorer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
|- VtxDeg = ( g e. _V |-> [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) ) )
Theorem	vtxdgfval 16409*	The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   =>    |- ( G e. W -> (VtxDeg ` G ) = ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) )
Theorem	vtxedgfi 16410*	In a finite graph, the number of edges from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> { x e. A | U e. ( I ` x ) } e. Fin )
Theorem	vtxlpfi 16411*	In a finite graph, the number of loops from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> { x e. A | ( I ` x ) = { U } } e. Fin )
Theorem	vtxdgfifival 16412*	The degree of a vertex for graphs with finite vertex and edge sets. (Contributed by Jim Kingdon, 10-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> ( (VtxDeg ` G ) ` U ) = ( (♯ ` { x e. A | U e. ( I ` x ) } ) + (♯ ` { x e. A | ( I ` x ) = { U } } ) ) )
Theorem	vtxdgop 16413	The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
|- ( G e. W -> (VtxDeg ` G ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )
Theorem	vtxdgfif 16414	In a finite graph, the vertex degree function is a function from vertices to nonnegative integers. (Contributed by Jim Kingdon, 17-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> (VtxDeg ` G ) : V --> NN0 )
Theorem	vtxdg0v 16415	The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.)
|- V = (Vtx ` G )   =>    |- ( ( G = (/) /\ U e. V ) -> ( (VtxDeg ` G ) ` U ) = 0 )
Theorem	vtxdgfi0e 16416	The degree of a vertex in an empty graph is zero, because there are no edges. This is the base case for the induction for calculating the degree of a vertex, for example in a Königsberg graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> U e. V )   &    |- ( ph -> I = (/) )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> ( (VtxDeg ` G ) ` U ) = 0 )
Theorem	vtxdeqd 16417	Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.)
|- ( ph -> G e. X )   &    |- ( ph -> H e. Y )   &    |- ( ph -> (Vtx ` H ) = (Vtx ` G ) )   &    |- ( ph -> (iEdg ` H ) = (iEdg ` G ) )   =>    |- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )
Theorem	vtxdfifiun 16418	The degree of a vertex in the union of two pseudographs of finite size on the same finite vertex set is the sum of the degrees of the vertex in each pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.)
|- I = (iEdg ` G )   &    |- J = (iEdg ` H )   &    |- V = (Vtx ` G )   &    |- ( ph -> (Vtx ` H ) = V )   &    |- ( ph -> (Vtx ` U ) = V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> H e. UPGraph )   &    |- ( ph -> ( dom I i^i dom J ) = (/) )   &    |- ( ph -> Fun I )   &    |- ( ph -> Fun J )   &    |- ( ph -> N e. V )   &    |- ( ph -> (iEdg ` U ) = ( I u. J ) )   &    |- ( ph -> dom I e. Fin )   &    |- ( ph -> dom J e. Fin )   =>    |- ( ph -> ( (VtxDeg ` U ) ` N ) = ( ( (VtxDeg ` G ) ` N ) + ( (VtxDeg ` H ) ` N ) ) )
Theorem	vtxdumgrfival 16419*	The value of the vertex degree function for a finite multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 23-Feb-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- D = (VtxDeg ` G )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> U e. V )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( D ` U ) = (♯ ` { x e. A | U e. ( I ` x ) } ) )
Theorem	vtxd0nedgbfi 16420*	A vertex has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 22-Mar-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- D = (VtxDeg ` G )   &    |- ( ph -> dom I e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> ( ( D ` U ) = 0 <-> -. E. i e. dom I U e. ( I ` i ) ) )
Theorem	vtxduspgrfvedgfilem 16421*	Lemma for vtxduspgrfvedgfi 16422 and vtxdusgrfvedgfi 16423. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   &    |- ( ph -> dom (iEdg ` G ) e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. USPGraph )   =>    |- ( ph -> (♯ ` { i e. dom (iEdg ` G ) | U e. ( (iEdg ` G ) ` i ) } ) = (♯ ` { e e. E | U e. e } ) )
Theorem	vtxduspgrfvedgfi 16422*	The value of the vertex degree function for a simple pseudograph. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   &    |- ( ph -> dom (iEdg ` G ) e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. USPGraph )   &    |- D = (VtxDeg ` G )   =>    |- ( ph -> ( D ` U ) = ( (♯ ` { e e. E | U e. e } ) + (♯ ` { e e. E | e = { U } } ) ) )
Theorem	vtxdusgrfvedgfi 16423*	The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.)
|- V = (Vtx ` G )   &    |- E = (Edg ` G )   &    |- ( ph -> dom (iEdg ` G ) e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. USGraph )   &    |- D = (VtxDeg ` G )   =>    |- ( ph -> ( D ` U ) = (♯ ` { e e. E | U e. e } ) )
Theorem	1loopgruspgr 16424	A graph with one edge which is a loop is a simple pseudograph. (Contributed by AV, 21-Feb-2021.)
|- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> N e. V )   &    |- ( ph -> (iEdg ` G ) = { <. A , { N } >. } )   =>    |- ( ph -> G e. USPGraph )
Theorem	1loopgredg 16425	The set of edges in a graph (simple pseudograph) with one edge which is a loop is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
|- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> N e. V )   &    |- ( ph -> (iEdg ` G ) = { <. A , { N } >. } )   =>    |- ( ph -> (Edg ` G ) = { { N } } )
Theorem	1loopgrvd2fi 16426	The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
|- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> N e. V )   &    |- ( ph -> (iEdg ` G ) = { <. A , { N } >. } )   &    |- ( ph -> V e. Fin )   =>    |- ( ph -> ( (VtxDeg ` G ) ` N ) = 2 )
Theorem	1loopgrvd0fi 16427	The vertex degree of a one-edge graph, case 1 (for a loop): a loop at a vertex other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 21-Feb-2021.)
|- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> N e. V )   &    |- ( ph -> (iEdg ` G ) = { <. A , { N } >. } )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> K e. ( V \ { N } ) )   =>    |- ( ph -> ( (VtxDeg ` G ) ` K ) = 0 )
Theorem	1hevtxdg0fi 16428	The vertex degree of vertex D in a finite pseudograph G with only one edge E is 0 if D is not incident with the edge E. (Contributed by AV, 2-Mar-2021.) (Revised by Jim Kingdon, 13-Mar-2026.)
|- ( ph -> (iEdg ` G ) = { <. A , E >. } )   &    |- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> D e. V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> E e. Y )   &    |- ( ph -> D e/ E )   =>    |- ( ph -> ( (VtxDeg ` G ) ` D ) = 0 )
Theorem	1hevtxdg1en 16429	The vertex degree of vertex D in a multigraph G with only one edge E is 1 if D is incident with the edge E. (Contributed by AV, 2-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
|- ( ph -> (iEdg ` G ) = { <. A , E >. } )   &    |- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> D e. V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UMGraph )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> D e. E )   &    |- ( ph -> E ~~ 2o )   =>    |- ( ph -> ( (VtxDeg ` G ) ` D ) = 1 )
Theorem	1hegrvtxdg1fi 16430	The vertex degree of a multigraph with one edge, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. V )   &    |- ( ph -> C e. V )   &    |- ( ph -> B =/= C )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> (iEdg ` G ) = { <. A , E >. } )   &    |- ( ph -> { B , C } C_ E )   &    |- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UMGraph )   =>    |- ( ph -> ( (VtxDeg ` G ) ` B ) = 1 )
Theorem	1hegrvtxdg1rfi 16431	The vertex degree of a graph with one hyperedge, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
|- ( ph -> A e. X )   &    |- ( ph -> B e. V )   &    |- ( ph -> C e. V )   &    |- ( ph -> B =/= C )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> (iEdg ` G ) = { <. A , E >. } )   &    |- ( ph -> { B , C } C_ E )   &    |- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UMGraph )   =>    |- ( ph -> ( (VtxDeg ` G ) ` C ) = 1 )
Theorem	p1evtxdeqfilem 16432	Lemma for p1evtxdeqfi 16433 and p1evtxdp1fi 16434. (Contributed by AV, 3-Mar-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> (Vtx ` F ) = V )   &    |- ( ph -> (iEdg ` F ) = ( I u. { <. K , E >. } ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> K e/ dom I )   &    |- ( ph -> U e. V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> dom I e. Fin )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   &    |- ( ph -> E e. Y )   =>    |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + ( (VtxDeg ` <. V , { <. K , E >. } >. ) ` U ) ) )
Theorem	p1evtxdeqfi 16433	If an edge E which does not contain vertex U is added to a graph G (yielding a graph F), the degree of U is the same in both graphs. (Contributed by AV, 2-Mar-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> (Vtx ` F ) = V )   &    |- ( ph -> (iEdg ` F ) = ( I u. { <. K , E >. } ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> K e/ dom I )   &    |- ( ph -> U e. V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> dom I e. Fin )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   &    |- ( ph -> E e. Y )   &    |- ( ph -> U e/ E )   =>    |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( (VtxDeg ` G ) ` U ) )
Theorem	p1evtxdp1fi 16434	If an edge E (not being a loop) which contains vertex U is added to a graph G (yielding a graph F), the degree of U is increased by 1. (Contributed by AV, 3-Mar-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> (Vtx ` F ) = V )   &    |- ( ph -> (iEdg ` F ) = ( I u. { <. K , E >. } ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> K e/ dom I )   &    |- ( ph -> U e. V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> dom I e. Fin )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   &    |- ( ph -> U e. E )   =>    |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( ( (VtxDeg ` G ) ` U ) + 1 ) )
Theorem	vdegp1aid 16435*	The induction step for a vertex degree calculation. If the degree of U in the edge set E is P, then adding { X , Y } to the edge set, where X =/= U =/= Y, yields degree P as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
|- V = (Vtx ` G )   &    |- ( ph -> U e. V )   &    |- I = (iEdg ` G )   &    |- ( ph -> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )   &    |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )   &    |- ( ph -> (Vtx ` F ) = V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> X e. V )   &    |- ( ph -> X =/= U )   &    |- ( ph -> Y e. V )   &    |- ( ph -> Y =/= U )   &    |- ( ph -> X =/= Y )   &    |- ( ph -> (iEdg ` F ) = ( I ++ <" { X , Y } "> ) )   =>    |- ( ph -> ( (VtxDeg ` F ) ` U ) = P )
Theorem	vdegp1bid 16436*	The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of U in the edge set E is P, then adding { U , X } to the edge set, where X =/= U, yields degree P + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
|- V = (Vtx ` G )   &    |- ( ph -> U e. V )   &    |- I = (iEdg ` G )   &    |- ( ph -> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )   &    |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )   &    |- ( ph -> (Vtx ` F ) = V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> X e. V )   &    |- ( ph -> X =/= U )   &    |- ( ph -> (iEdg ` F ) = ( I ++ <" { U , X } "> ) )   =>    |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( P + 1 ) )
Theorem	vdegp1cid 16437*	The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of U in the edge set E is P, then adding { X , U } to the edge set, where X =/= U, yields degree P + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
|- V = (Vtx ` G )   &    |- ( ph -> U e. V )   &    |- I = (iEdg ` G )   &    |- ( ph -> I e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )   &    |- ( ph -> ( (VtxDeg ` G ) ` U ) = P )   &    |- ( ph -> (Vtx ` F ) = V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> X e. V )   &    |- ( ph -> X =/= U )   &    |- ( ph -> (iEdg ` F ) = ( I ++ <" { X , U } "> ) )   =>    |- ( ph -> ( (VtxDeg ` F ) ` U ) = ( P + 1 ) )
12.3  Walks, paths and cycles
12.3.1  Walks
Syntax	cwlks 16438	Extend class notation with walks (i.e. 1-walks) (of a hypergraph).
class Walks
Definition	df-wlks 16439*	Define the set of all walks (in a hypergraph). Such walks correspond to the s-walks "on the vertex level" (with s = 1), and also to 1-walks "on the edge level" (see wlk1walkdom 16480) discussed in Aksoy et al. The predicate F (Walks ` G ) P can be read as "The pair <. F , P >. represents a walk in a graph G", see also iswlk 16444. The condition { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. ( p ` k ) = ( p ` ( k + 1 ) ) should be allowed only if there is a loop at ( p ` k ). Otherwise, C would be fulfilled by each edge containing ( p ` k ). According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is 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 AV, 30-Dec-2020.)
|- Walks = ( g e. _V |-> { <. f , p >. | ( f e. Word dom (iEdg ` g ) /\ p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) ) ) } )
Theorem	wlkmex 16440	If there are walks on a graph, the graph is a set. (Contributed by Jim Kingdon, 1-Feb-2026.)
|- ( W e. (Walks ` G ) -> G e. _V )
Theorem	wkslem1 16441	Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)
|- ( A = B -> (if- ( ( P ` A ) = ( P ` ( A + 1 ) ) , ( I ` ( F ` A ) ) = { ( P ` A ) } , { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) ) <-> if- ( ( P ` B ) = ( P ` ( B + 1 ) ) , ( I ` ( F ` B ) ) = { ( P ` B ) } , { ( P ` B ) , ( P ` ( B + 1 ) ) } C_ ( I ` ( F ` B ) ) ) ) )
Theorem	wkslem2 16442	Lemma 2 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)
|- ( ( A = B /\ ( A + 1 ) = C ) -> (if- ( ( P ` A ) = ( P ` ( A + 1 ) ) , ( I ` ( F ` A ) ) = { ( P ` A ) } , { ( P ` A ) , ( P ` ( A + 1 ) ) } C_ ( I ` ( F ` A ) ) ) <-> if- ( ( P ` B ) = ( P ` C ) , ( I ` ( F ` B ) ) = { ( P ` B ) } , { ( P ` B ) , ( P ` C ) } C_ ( I ` ( F ` B ) ) ) ) )
Theorem	wksfval 16443*	The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( G e. W -> (Walks ` G ) = { <. f , p >. | ( f e. Word dom I /\ p : ( 0 ... (♯ ` f ) ) --> V /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( I ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( I ` ( f ` k ) ) ) ) } )
Theorem	iswlk 16444*	Properties of a pair of functions to be/represent a walk. (Contributed by AV, 30-Dec-2020.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ( G e. W /\ F e. U /\ P e. Z ) -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
Theorem	wlkpropg 16445*	Properties of a walk. (Contributed by AV, 5-Nov-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
Theorem	wlkex 16446	The class of walks on a graph is a set. (Contributed by Jim Kingdon, 7-Feb-2026.)
|- ( G e. V -> (Walks ` G ) e. _V )
Theorem	wlkv 16447	The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.)
|- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
Theorem	wlkprop 16448*	Properties of a walk. (Contributed by AV, 5-Nov-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( F (Walks ` G ) P -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
Theorem	wlkvg 16449	The classes involved in a walk are sets. Now that we have wlkv 16447 there is no reason to use this theorem in new proofs and using wlkv 16447 is encouraged for consistency with the Metamath Proof Explorer. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.) (New usage is discouraged.)
|- ( ( G e. W /\ F (Walks ` G ) P ) -> ( F e. _V /\ P e. _V ) )
Theorem	iswlkg 16450*	Generalization of iswlk 16444: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   =>    |- ( G e. W -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
Theorem	wlkf 16451	The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)
|- I = (iEdg ` G )   =>    |- ( F (Walks ` G ) P -> F e. Word dom I )
Theorem	wlkfg 16452	The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> F e. Word dom I )
Theorem	wlkcl 16453	A walk has length ♯ ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
|- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
Theorem	wlkclg 16454	A walk has length ♯ ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
|- ( ( G e. W /\ F (Walks ` G ) P ) -> (♯ ` F ) e. NN0 )
Theorem	wlkp 16455	The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)
|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
Theorem	wlkpg 16456	The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)
|- V = (Vtx ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> P : ( 0 ... (♯ ` F ) ) --> V )
Theorem	wlkpwrdg 16457	The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.)
|- V = (Vtx ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> P e. Word V )
Theorem	wlklenvp1 16458	The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.)
|- ( F (Walks ` G ) P -> (♯ ` P ) = ( (♯ ` F ) + 1 ) )
Theorem	wlklenvp1g 16459	The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.)
|- ( ( G e. W /\ F (Walks ` G ) P ) -> (♯ ` P ) = ( (♯ ` F ) + 1 ) )
Theorem	wlkm 16460*	The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.)
|- ( F (Walks ` G ) P -> E. x x e. P )
Theorem	wlkvtxm 16461*	A graph with a walk has at least one vertex. (Contributed by Jim Kingdon, 8-Feb-2026.)
|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> E. x x e. V )
Theorem	wlklenvm1 16462	The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.)
|- ( F (Walks ` G ) P -> (♯ ` F ) = ( (♯ ` P ) - 1 ) )
Theorem	wlklenvm1g 16463	The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.)
|- ( ( G e. W /\ F (Walks ` G ) P ) -> (♯ ` F ) = ( (♯ ` P ) - 1 ) )
Theorem	ifpsnprss 16464	Lemma for wlkvtxeledgg 16465: Two adjacent (not necessarily different) vertices A and B in a walk are incident with an edge E. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.)
|- (if- ( A = B , E = { A } , { A , B } C_ E ) -> { A , B } C_ E )
Theorem	wlkvtxeledgg 16465*	Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
Theorem	wlkvtxiedg 16466*	The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
|- I = (iEdg ` G )   =>    |- ( F (Walks ` G ) P -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
Theorem	wlkvtxiedgg 16467*	The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
|- I = (iEdg ` G )   =>    |- ( ( G e. W /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
Theorem	relwlk 16468	The set (Walks ` G ) of all walks on G is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.)
|- Rel (Walks ` G )
Theorem	wlkop 16469	A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.)
|- ( W e. (Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
Theorem	wlkelvv 16470	A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)
|- ( W e. (Walks ` G ) -> W e. ( _V X. _V ) )
Theorem	wlkcprim 16471	A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Revised by Jim Kingdon, 1-Feb-2026.)
|- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
Theorem	wlk2f 16472*	If there is a walk W there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
|- ( W e. (Walks ` G ) -> E. f E. p f (Walks ` G ) p )
Theorem	wlkcompim 16473*	Implications for the properties of the components of a walk. (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 e. (Walks ` G ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
Theorem	wlkelwrd 16474	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 e. (Walks ` G ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V ) )
Theorem	wlkeq 16475*	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 e. (Walks ` G ) /\ B e. (Walks ` G ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. x e. ( 0..^ N ) ( ( 1st ` A ) ` x ) = ( ( 1st ` B ) ` x ) /\ A. x e. ( 0 ... N ) ( ( 2nd ` A ) ` x ) = ( ( 2nd ` B ) ` x ) ) ) )
Theorem	edginwlkd 16476	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.) (Revised by Jim Kingdon, 2-Feb-2026.)
|- I = (iEdg ` G )   &    |- E = (Edg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> F e. Word dom I )   &    |- ( ph -> K e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> G e. V )   =>    |- ( ph -> ( I ` ( F ` K ) ) e. E )
Theorem	upgredginwlk 16477	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 e. UPGraph /\ F e. Word dom I ) -> ( K e. ( 0..^ (♯ ` F ) ) -> ( I ` ( F ` K ) ) e. E ) )
Theorem	iedginwlk 16478	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 e. ( 0..^ (♯ ` F ) ) ) -> ( I ` ( F ` X ) ) e. ran I )
Theorem	wlkl1loop 16479	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 ) } e. (Edg ` G ) )
Theorem	wlk1walkdom 16480*	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 -> A. k e. ( 1..^ (♯ ` F ) ) 1o ~<_ ( ( I ` ( F ` ( k - 1 ) ) ) i^i ( I ` ( F ` k ) ) ) )
Theorem	upgriswlkdc 16481*	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 e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) (DECID ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) )
Theorem	upgrwlkedg 16482*	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 e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
Theorem	upgrwlkcompim 16483*	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 e. UPGraph /\ W e. (Walks ` G ) ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
Theorem	wlkvtxedg 16484*	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 -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. E { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
Theorem	upgrwlkvtxedg 16485*	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 e. UPGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } e. E )
Theorem	uspgr2wlkeq 16486*	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 e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ N = (♯ ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = (♯ ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) )
Theorem	uspgr2wlkeq2 16487	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 e. USPGraph /\ N e. NN0 ) /\ ( A e. (Walks ` G ) /\ (♯ ` ( 1st ` A ) ) = N ) /\ ( B e. (Walks ` G ) /\ (♯ ` ( 1st ` B ) ) = N ) ) -> ( ( 2nd ` A ) = ( 2nd ` B ) -> A = B ) )
Theorem	uspgr2wlkeqi 16488	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 e. USPGraph /\ ( A e. (Walks ` G ) /\ B e. (Walks ` G ) ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) -> A = B )
Theorem	umgrwlknloop 16489*	In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 3-Jan-2021.)
|- ( ( G e. UMGraph /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )
Theorem	wlkv0 16490	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 e. (Walks ` G ) ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) )
Theorem	g0wlk0 16491	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 16492	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 16493	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 e/ _V /\ (Vtx ` S ) = (Vtx ` G ) ) -> (Walks ` G ) = (/) )
Theorem	wlklenvclwlk 16494	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 e. Word (Vtx ` G ) -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) -> (♯ ` F ) = (♯ ` W ) ) )
Theorem	wlkpvtx 16495	A walk connects vertices. (Contributed by AV, 22-Feb-2021.)
|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> ( N e. ( 0 ... (♯ ` F ) ) -> ( P ` N ) e. V ) )
Theorem	wlkepvtx 16496	The endpoints of a walk are vertices. (Contributed by AV, 31-Jan-2021.)
|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> ( ( P ` 0 ) e. V /\ ( P ` (♯ ` F ) ) e. V ) )
Theorem	2wlklem 16497*	Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
|- ( A. k e. { 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	upgr2wlkdc 16498*	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 e. UPGraph -> ( ( F (Walks ` G ) P /\ F ~~ 2o ) <-> ( ( F : ( 0..^ 2 ) --> dom I /\ P : ( 0 ... 2 ) --> V /\ A. k e. { 0 , 1 }DECID ( P ` k ) = ( P ` ( k + 1 ) ) ) /\ ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
Theorem	wlkreslem 16499	Lemma for wlkres 16500. (Contributed by AV, 5-Mar-2021.) (Revised by AV, 30-Nov-2022.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- ( ph -> F (Walks ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> (Vtx ` S ) = V )   =>    |- ( ph -> S e. _V )
Theorem	wlkres 16500	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. (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 )   &    |- ( ph -> F (Walks ` G ) P )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> (Vtx ` S ) = V )   &    |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )   &    |- H = ( F prefix N )   &    |- Q = ( P |` ( 0 ... N ) )   =>    |- ( ph -> H (Walks ` S ) Q )