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	usgr2v1e2w 16401	A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
|- ( ( A e. X /\ B e. Y /\ A =/= B ) -> <. { A , B } , <" { A , B } "> >. e. USGraph )
Theorem	edg0usgr 16402	A class without edges is a simple graph. Since ran F = (/) does not generally imply Fun F, but Fun (iEdg ` G ) is required for G to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)
|- ( ( G e. W /\ (Edg ` G ) = (/) /\ Fun (iEdg ` G ) ) -> G e. USGraph )
Theorem	usgr1vr 16403	A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 2-Apr-2021.)
|- ( ( A e. X /\ (Vtx ` G ) = { A } ) -> ( G e. USGraph -> (iEdg ` G ) = (/) ) )
Theorem	usgrexmpldifpr 16404	Lemma for usgrexmpledg : all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
|- ( ( { 0 , 1 } =/= { 1 , 2 } /\ { 0 , 1 } =/= { 2 , 0 } /\ { 0 , 1 } =/= { 0 , 3 } ) /\ ( { 1 , 2 } =/= { 2 , 0 } /\ { 1 , 2 } =/= { 0 , 3 } /\ { 2 , 0 } =/= { 0 , 3 } ) )
Theorem	griedg0prc 16405*	The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.)
|- U = { <. v , e >. | e : (/) --> (/) }   =>    |- U e/ _V
Theorem	griedg0ssusgr 16406*	The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.)
|- U = { <. v , e >. | e : (/) --> (/) }   =>    |- U C_ USGraph
Theorem	usgrprc 16407	The class of simple graphs is a proper class (and therefore, because of prcssprc 4269, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)
|- USGraph e/ _V
12.2.7  Subgraphs
Syntax	csubgr 16408	Extend class notation with subgraphs.
class SubGraph
Definition	df-subgr 16409*	Define the class of the subgraph relation. A class s is a subgraph of a class g (the supergraph of s) if its vertices are also vertices of g, and its edges are also edges of g, connecting vertices of s only (see section I.1 in [Bollobas] p. 2 or section 1.1 in [Diestel] p. 4). The second condition is ensured by the requirement that the edge function of s is a restriction of the edge function of g having only vertices of s in its range. Note that the domains of the edge functions of the subgraph and the supergraph should be compatible. (Contributed by AV, 16-Nov-2020.)
|- SubGraph = { <. s , g >. | ( (Vtx ` s ) C_ (Vtx ` g ) /\ (iEdg ` s ) = ( (iEdg ` g ) |` dom (iEdg ` s ) ) /\ (Edg ` s ) C_ ~P (Vtx ` s ) ) }
Theorem	relsubgr 16410	The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.)
|- Rel SubGraph
Theorem	subgrv 16411	If a class is a subgraph of another class, both classes are sets. (Contributed by AV, 16-Nov-2020.)
|- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
Theorem	issubgr 16412	The property of a set to be a subgraph of another set. (Contributed by AV, 16-Nov-2020.)
|- V = (Vtx ` S )   &    |- A = (Vtx ` G )   &    |- I = (iEdg ` S )   &    |- B = (iEdg ` G )   &    |- E = (Edg ` S )   =>    |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
Theorem	issubgr2 16413	The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-Nov-2020.)
|- V = (Vtx ` S )   &    |- A = (Vtx ` G )   &    |- I = (iEdg ` S )   &    |- B = (iEdg ` G )   &    |- E = (Edg ` S )   =>    |- ( ( G e. W /\ Fun B /\ S e. U ) -> ( S SubGraph G <-> ( V C_ A /\ I C_ B /\ E C_ ~P V ) ) )
Theorem	subgrprop 16414	The properties of a subgraph. (Contributed by AV, 19-Nov-2020.)
|- V = (Vtx ` S )   &    |- A = (Vtx ` G )   &    |- I = (iEdg ` S )   &    |- B = (iEdg ` G )   &    |- E = (Edg ` S )   =>    |- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )
Theorem	subgrprop2 16415	The properties of a subgraph: If S is a subgraph of G, its vertices are also vertices of G, and its edges are also edges of G, connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020.)
|- V = (Vtx ` S )   &    |- A = (Vtx ` G )   &    |- I = (iEdg ` S )   &    |- B = (iEdg ` G )   &    |- E = (Edg ` S )   =>    |- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ E C_ ~P V ) )
Theorem	uhgrissubgr 16416	The property of a hypergraph to be a subgraph. (Contributed by AV, 19-Nov-2020.)
|- V = (Vtx ` S )   &    |- A = (Vtx ` G )   &    |- I = (iEdg ` S )   &    |- B = (iEdg ` G )   =>    |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> ( S SubGraph G <-> ( V C_ A /\ I C_ B ) ) )
Theorem	subgrprop3 16417	The properties of a subgraph: If S is a subgraph of G, its vertices are also vertices of G, and its edges are also edges of G. (Contributed by AV, 19-Nov-2020.)
|- V = (Vtx ` S )   &    |- A = (Vtx ` G )   &    |- E = (Edg ` S )   &    |- B = (Edg ` G )   =>    |- ( S SubGraph G -> ( V C_ A /\ E C_ B ) )
Theorem	egrsubgr 16418	An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 17-Dec-2020.)
|- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> S SubGraph G )
Theorem	0grsubgr 16419	The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
|- ( G e. W -> (/) SubGraph G )
Theorem	0uhgrsubgr 16420	The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.)
|- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> S SubGraph G )
Theorem	uhgrsubgrself 16421	A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
|- ( G e. UHGraph -> G SubGraph G )
Theorem	subgrfun 16422	The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.)
|- ( ( Fun (iEdg ` G ) /\ S SubGraph G ) -> Fun (iEdg ` S ) )
Theorem	subgruhgrfun 16423	The edge function of a subgraph of a hypergraph is a function. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 20-Nov-2020.)
|- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
Theorem	subgreldmiedg 16424	An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.)
|- ( ( S SubGraph G /\ X e. dom (iEdg ` S ) ) -> X e. dom (iEdg ` G ) )
Theorem	subgruhgredgdm 16425*	An edge of a subgraph of a hypergraph is an inhabited subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)
|- V = (Vtx ` S )   &    |- I = (iEdg ` S )   &    |- ( ph -> G e. UHGraph )   &    |- ( ph -> S SubGraph G )   &    |- ( ph -> X e. dom I )   =>    |- ( ph -> ( I ` X ) e. { s e. ~P V | E. j j e. s } )
Theorem	subumgredg2en 16426*	An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020.)
|- V = (Vtx ` S )   &    |- I = (iEdg ` S )   =>    |- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. { e e. ~P V | e ~~ 2o } )
Theorem	subuhgr 16427	A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
|- ( ( G e. UHGraph /\ S SubGraph G ) -> S e. UHGraph )
Theorem	subupgr 16428	A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
|- ( ( G e. UPGraph /\ S SubGraph G ) -> S e. UPGraph )
Theorem	subumgr 16429	A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.)
|- ( ( G e. UMGraph /\ S SubGraph G ) -> S e. UMGraph )
Theorem	subusgr 16430	A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.)
|- ( ( G e. USGraph /\ S SubGraph G ) -> S e. USGraph )
Theorem	uhgrspansubgrlem 16431	Lemma for uhgrspansubgr 16432: The edges of the graph S obtained by removing some edges of a hypergraph G are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 16432. (Contributed 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. UHGraph )   =>    |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )
Theorem	uhgrspansubgr 16432	A spanning subgraph S of a hypergraph G is actually a subgraph of G. A subgraph S of a graph G which has the same vertices as G and is obtained by removing some edges of G is called a spanning subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). Formally, the edges are "removed" by restricting the edge function of the original graph by an arbitrary class (which actually needs not to be a subset of the domain of the edge function). (Contributed by AV, 18-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
|- V = (Vtx ` G )   &    |- E = (iEdg ` G )   &    |- ( ph -> S e. W )   &    |- ( ph -> (Vtx ` S ) = V )   &    |- ( ph -> (iEdg ` S ) = ( E |` A ) )   &    |- ( ph -> G e. UHGraph )   =>    |- ( ph -> S SubGraph G )
Theorem	uhgrspan 16433	A spanning subgraph S of a hypergraph G is a hypergraph. (Contributed by AV, 11-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. UHGraph )   =>    |- ( ph -> S e. UHGraph )
Theorem	upgrspan 16434	A spanning subgraph S of a pseudograph G is a pseudograph. (Contributed by AV, 11-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. UPGraph )   =>    |- ( ph -> S e. UPGraph )
Theorem	umgrspan 16435	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 16436	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 16437	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 16438	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 16439	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 16440	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 16441	Extend class notation with the vertex degree function.
class VtxDeg
Definition	df-vtxdg 16442*	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 7203), 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 16443*	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 16444*	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 16445*	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 16446*	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 16447	The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
|- ( G e. W -> (VtxDeg ` G ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )
Theorem	vtxdgfif 16448	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 16449	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 16450	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 16451	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 16452	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 16453*	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 16454*	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 16455*	Lemma for vtxduspgrfvedgfi 16456 and vtxdusgrfvedgfi 16457. (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 16456*	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 16457*	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 16458	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 16459	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 16460	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 16461	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 16462	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 16463	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 16464	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 16465	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 16466	Lemma for p1evtxdeqfi 16467 and p1evtxdp1fi 16468. (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 16467	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 16468	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 16469*	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 16470*	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 16471*	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 16472	Extend class notation with walks (i.e. 1-walks) (of a hypergraph).
class Walks
Definition	df-wlks 16473*	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 16514) 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 16478. 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 16474	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 16475	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 16476	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 16477*	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 16478*	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 16479*	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 16480	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 16481	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 16482*	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 16483	The classes involved in a walk are sets. Now that we have wlkv 16481 there is no reason to use this theorem in new proofs and using wlkv 16481 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 16484*	Generalization of iswlk 16478: 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 16485	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 16486	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 16487	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 16488	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 16489	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 16490	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 16491	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 16492	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 16493	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 16494*	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 16495*	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 16496	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 16497	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 16498	Lemma for wlkvtxeledgg 16499: 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 16499*	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 16500*	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 )