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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuhgr2edg 27001* If a vertex is adjacent to two different vertices in a hypergraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremumgr2edg 27002* If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremusgr2edg 27003* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))

Theoremumgr2edg1 27004* If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 8-Jun-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼𝑥))

Theoremusgr2edg1 27005* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 8-Jun-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ USGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥 ∈ dom 𝐼 𝑁 ∈ (𝐼𝑥))

Theoremumgrvad2edg 27006* If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex, analogous to usgr2edg 27003. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)
𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥𝐸𝑦𝐸 (𝑥𝑦𝑁𝑥𝑁𝑦))

Theoremumgr2edgneu 27007* If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex, analogous to usgr2edg1 27005. Lemma for theorems about friendship graphs. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UMGraph ∧ 𝐴𝐵) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ¬ ∃!𝑥𝐸 𝑁𝑥)

Theoremusgrsizedg 27008 In a simple graph, the size of the edge function is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 7-Jun-2021.)
(𝐺 ∈ USGraph → (♯‘(iEdg‘𝐺)) = (♯‘(Edg‘𝐺)))

Theoremusgredg3 27009* The value of the "edge function" of a simple graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ∃𝑥𝑉𝑦𝑉 (𝑥𝑦 ∧ (𝐸𝑋) = {𝑥, 𝑦}))

Theoremusgredg4 27010* For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})

Theoremusgredgreu 27011* For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸𝑌 ∈ (𝐸𝑋)) → ∃!𝑦𝑉 (𝐸𝑋) = {𝑌, 𝑦})

Theoremusgredg2vtx 27012* For a vertex incident to an edge there is another vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 5-Dec-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremuspgredg2vtxeu 27013* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 6-Dec-2020.)
((𝐺 ∈ USPGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremusgredg2vtxeu 27014* For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremusgredg2vtxeuALT 27015* Alternate proof of usgredg2vtxeu 27014, using edgiedgb 26850, the general translation from (iEdg‘𝐺) to (Edg‘𝐺). (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Theoremuspgredg2vlem 27016* Lemma for uspgredg2v 27017. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐴 = {𝑒𝐸𝑁𝑒}       ((𝐺 ∈ USPGraph ∧ 𝑌𝐴) → (𝑧𝑉 𝑌 = {𝑁, 𝑧}) ∈ 𝑉)

Theoremuspgredg2v 27017* In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐴 = {𝑒𝐸𝑁𝑒}    &   𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 𝑦 = {𝑁, 𝑧}))       ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)

Theoremusgredg2vlem1 27018* Lemma 1 for usgredg2v 27020. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}       ((𝐺 ∈ USGraph ∧ 𝑌𝐴) → (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉)

Theoremusgredg2vlem2 27019* Lemma 2 for usgredg2v 27020. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}       ((𝐺 ∈ USGraph ∧ 𝑌𝐴) → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁}))

Theoremusgredg2v 27020* In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}    &   𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}))       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)

Theoremusgriedgleord 27021* Alternate version of usgredgleord 27026, not using the notation (Edg‘𝐺). In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (♯‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) ≤ (♯‘𝑉))

Theoremushgredgedg 27022* In a simple hypergraph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 11-Dec-2020.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)    &   𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}    &   𝐵 = {𝑒𝐸𝑁𝑒}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)

Theoremusgredgedg 27023* In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)    &   𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}    &   𝐵 = {𝑒𝐸𝑁𝑒}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)

Theoremushgredgedgloop 27024* In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex 𝑁 and the set of loops at this vertex 𝑁. (Contributed by AV, 11-Dec-2020.) (Revised by AV, 6-Jul-2022.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}    &   𝐵 = {𝑒𝐸𝑒 = {𝑁}}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)

Theoremuspgredgleord 27025* In a simple pseudograph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) ≤ (♯‘𝑉))

Theoremusgredgleord 27026* In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) ≤ (♯‘𝑉))

TheoremusgredgleordALT 27027* Alternate proof for usgredgleord 27026 based on usgriedgleord 27021. In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 5-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) ≤ (♯‘𝑉))

Theoremusgrstrrepe 27028* Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.)
𝑉 = (Base‘𝐺)    &   𝐼 = (.ef‘ndx)    &   (𝜑𝐺 Struct 𝑋)    &   (𝜑 → (Base‘ndx) ∈ dom 𝐺)    &   (𝜑𝐸𝑊)    &   (𝜑𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})       (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph)

16.2.6  Examples for graphs

Theoremusgr0e 27029 The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.)
(𝜑𝐺𝑊)    &   (𝜑 → (iEdg‘𝐺) = ∅)       (𝜑𝐺 ∈ USGraph)

Theoremusgr0vb 27030 The null graph, with no vertices, is a simple graph iff the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 16-Oct-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))

Theoremuhgr0v0e 27031 The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅)

Theoremuhgr0vsize0 27032 The size of a hypergraph with no vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = 0)

Theoremuhgr0edgfi 27033 A graph of order 0 (i.e. with 0 vertices) has a finite set of edges. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 8-Jun-2021.)
((𝐺 ∈ UHGraph ∧ (♯‘(Vtx‘𝐺)) = 0) → (Edg‘𝐺) ∈ Fin)

Theoremusgr0v 27034 The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph)

Theoremuhgr0vusgr 27035 The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.)
((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph)

Theoremusgr0 27036 The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
∅ ∈ USGraph

Theoremuspgr1e 27037 A simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})       (𝜑𝐺 ∈ USPGraph)

Theoremusgr1e 27038 A simple graph with one edge (with additional assumption that 𝐵𝐶 since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})    &   (𝜑𝐵𝐶)       (𝜑𝐺 ∈ USGraph)

Theoremusgr0eop 27039 The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
(𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ USGraph)

Theoremuspgr1eop 27040 A simple pseudograph with (at least) two vertices and one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ USPGraph)

Theoremuspgr1ewop 27041 A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
((𝑉𝑊𝐴𝑉𝐵𝑉) → ⟨𝑉, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USPGraph)

Theoremuspgr1v1eop 27042 A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.)
((𝑉𝑊𝐴𝑋𝐵𝑉) → ⟨𝑉, {⟨𝐴, {𝐵}⟩}⟩ ∈ USPGraph)

Theoremusgr1eop 27043 A simple graph with (at least) two different vertices and one edge. If the two vertices were not different, the edge would be a loop. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → (𝐵𝐶 → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ USGraph))

Theoremuspgr2v1e2w 27044 A simple pseudograph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
((𝐴𝑋𝐵𝑌) → ⟨{𝐴, 𝐵}, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USPGraph)

Theoremusgr2v1e2w 27045 A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.)
((𝐴𝑋𝐵𝑌𝐴𝐵) → ⟨{𝐴, 𝐵}, ⟨“{𝐴, 𝐵}”⟩⟩ ∈ USGraph)

Theoremedg0usgr 27046 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.)
((𝐺𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph)

Theoremlfuhgr1v0e 27047* A loop-free hypergraph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}       ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 1 ∧ 𝐼:dom 𝐼𝐸) → (Edg‘𝐺) = ∅)

Theoremusgr1vr 27048 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.)
((𝐴𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅))

Theoremusgr1v 27049 A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))

Theoremusgr1v0edg 27050 A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = {𝐴} ∧ Fun (iEdg‘𝐺)) → (𝐺 ∈ USGraph ↔ (Edg‘𝐺) = ∅))

Theoremusgrexmpldifpr 27051 Lemma for usgrexmpledg 27055: 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}))

Theoremusgrexmplef 27052* Lemma for usgrexmpl 27056. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩       𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}

Theoremusgrexmpllem 27053 Lemma for usgrexmpl 27056. (Contributed by AV, 21-Oct-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸)

Theoremusgrexmplvtx 27054 The vertices 0, 1, 2, 3, 4 of the graph 𝐺 = ⟨𝑉, 𝐸. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4})

Theoremusgrexmpledg 27055 The edges {0, 1}, {1, 2}, {2, 0}, {0, 3} of the graph 𝐺 = ⟨𝑉, 𝐸. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}})

Theoremusgrexmpl 27056 𝐺 is a simple graph of five vertices 0, 1, 2, 3, 4, with edges {0, 1}, {1, 2}, {2, 0}, {0, 3}. (Contributed by Alexander van der Vekens, 15-Aug-2017.) (Revised by AV, 21-Oct-2020.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       𝐺 ∈ USGraph

Theoremgriedg0prc 27057* The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.)
𝑈 = {⟨𝑣, 𝑒⟩ ∣ 𝑒:∅⟶∅}       𝑈 ∉ V

Theoremgriedg0ssusgr 27058* 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 ⊆ USGraph

Theoremusgrprc 27059 The class of simple graphs is a proper class (and therefore, because of prcssprc 5196, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)
USGraph ∉ V

16.2.7  Subgraphs

Syntaxcsubgr 27060 Extend class notation with subgraphs.
class SubGraph

Definitiondf-subgr 27061* 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) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))}

Theoremrelsubgr 27062 The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.)
Rel SubGraph

Theoremsubgrv 27063 If a class is a subgraph of another class, both classes are sets. (Contributed by AV, 16-Nov-2020.)
(𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))

Theoremissubgr 27064 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)       ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))

Theoremissubgr2 27065 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)       ((𝐺𝑊 ∧ Fun 𝐵𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉)))

Theoremsubgrprop 27066 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 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))

Theoremsubgrprop2 27067 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 → (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉))

Theoremuhgrissubgr 27068 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)       ((𝐺𝑊 ∧ Fun 𝐵𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼𝐵)))

Theoremsubgrprop3 27069 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 → (𝑉𝐴𝐸𝐵))

Theoremegrsubgr 27070 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.)
(((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)

Theorem0grsubgr 27071 The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
(𝐺𝑊 → ∅ SubGraph 𝐺)

Theorem0uhgrsubgr 27072 The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.)
((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)

Theoremuhgrsubgrself 27073 A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
(𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

Theoremsubgrfun 27074 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))

Theoremsubgruhgrfun 27075 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 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Theoremsubgreldmiedg 27076 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 𝐺𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺))

Theoremsubgruhgredgd 27077 An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)
𝑉 = (Vtx‘𝑆)    &   𝐼 = (iEdg‘𝑆)    &   (𝜑𝐺 ∈ UHGraph)    &   (𝜑𝑆 SubGraph 𝐺)    &   (𝜑𝑋 ∈ dom 𝐼)       (𝜑 → (𝐼𝑋) ∈ (𝒫 𝑉 ∖ {∅}))

Theoremsubumgredg2 27078* 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 𝐺𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2})

Theoremsubuhgr 27079 A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph)

Theoremsubupgr 27080 A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)

Theoremsubumgr 27081 A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.)
((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph)

Theoremsubusgr 27082 A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.)
((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph)

Theoremuhgrspansubgrlem 27083 Lemma for uhgrspansubgr 27084: 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 27084. (Contributed by AV, 18-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph)       (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))

Theoremuhgrspansubgr 27084 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)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph)       (𝜑𝑆 SubGraph 𝐺)

Theoremuhgrspan 27085 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)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UHGraph)       (𝜑𝑆 ∈ UHGraph)

Theoremupgrspan 27086 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)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UPGraph)       (𝜑𝑆 ∈ UPGraph)

Theoremumgrspan 27087 A spanning subgraph 𝑆 of a multigraph 𝐺 is a multigraph. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ UMGraph)       (𝜑𝑆 ∈ UMGraph)

Theoremusgrspan 27088 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)    &   (𝜑𝑆𝑊)    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))    &   (𝜑𝐺 ∈ USGraph)       (𝜑𝑆 ∈ USGraph)

Theoremuhgrspanop 27089 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 ∈ UHGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UHGraph)

Theoremupgrspanop 27090 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 ∈ UPGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UPGraph)

Theoremumgrspanop 27091 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 ∈ UMGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UMGraph)

Theoremusgrspanop 27092 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 ∈ USGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ USGraph)

Theoremuhgrspan1lem1 27093 Lemma 1 for uhgrspan1 27096. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}       ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)

Theoremuhgrspan1lem2 27094 Lemma 2 for uhgrspan1 27096. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       (Vtx‘𝑆) = (𝑉 ∖ {𝑁})

Theoremuhgrspan1lem3 27095 Lemma 3 for uhgrspan1 27096. (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       (iEdg‘𝑆) = (𝐼𝐹)

Theoremuhgrspan1 27096* The induced subgraph 𝑆 of a hypergraph 𝐺 obtained by removing one vertex is actually a subgraph of 𝐺. A subgraph is called induced or spanned by a subset of vertices of a graph if it contains all edges of the original graph that join two vertices of the subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). (Contributed by AV, 19-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩       ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 𝑆 SubGraph 𝐺)

Theoremupgrreslem 27097* Lemma for upgrres 27099. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})

Theoremumgrreslem 27098* Lemma for umgrres 27100 and usgrres 27101. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ran (𝐸𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2})

Theoremupgrres 27099* A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 27096) is a pseudograph. (Contributed by AV, 8-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩       ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)

Theoremumgrres 27100* A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 27096) is a multigraph. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸𝐹)⟩       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → 𝑆 ∈ UMGraph)

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