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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremusgrf1o 40401 The edge function of a simple graph is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1-onto→ran 𝐸)

Theoremusgrf1 40402 The edge function of a simple graph is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→ran 𝐸)

Theoremuspgrf1oedg 40403 The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))

Theoremusgrss 40404 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 15-Oct-2020.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸𝑋) ⊆ 𝑉)

Theoremuspgrushgr 40405 A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph )

Theoremuspgrupgr 40406 A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )

Theoremuspgrupgrushgr 40407 A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.)
(𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ))

Theoremusgruspgr 40408 A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )

Theoremusgrumgr 40409 A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )

Theoremusgrumgruspgr 40410 A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020.)
(𝐺 ∈ USGraph ↔ (𝐺 ∈ UMGraph ∧ 𝐺 ∈ USPGraph ))

Theoremusgruspgrb 40411* A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)
(𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2))

Theoremusgrupgr 40412 A simple graph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )

Theoremusgruhgr 40413 A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.)
(𝐺 ∈ USGraph → 𝐺 ∈ UHGraph )

Theoremusgrislfuspgr 40414* A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}))

Theoremuspgrun 40415 The union 𝑈 of two simple pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
(𝜑𝐺 ∈ USPGraph )    &   (𝜑𝐻 ∈ USPGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UPGraph )

Theoremuspgrunop 40416 The union of two simple pseudographs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple pseudographs, then 𝑉, 𝐸𝐹 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.)
(𝜑𝐺 ∈ USPGraph )    &   (𝜑𝐻 ∈ USPGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UPGraph )

Theoremusgrun 40417 The union 𝑈 of two simple graphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a multigraph (not necessarily a simple graph!) with the vertex 𝑉 and the union (𝐸𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.)
(𝜑𝐺 ∈ USGraph )    &   (𝜑𝐻 ∈ USGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)    &   (𝜑𝑈𝑊)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐸𝐹))       (𝜑𝑈 ∈ UMGraph )

Theoremusgrunop 40418 The union of two simple graphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are simple graphs, then 𝑉, 𝐸𝐹 is a multigraph (not necessarily a simple graph!) - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.)
(𝜑𝐺 ∈ USGraph )    &   (𝜑𝐻 ∈ USGraph )    &   𝐸 = (iEdg‘𝐺)    &   𝐹 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)       (𝜑 → ⟨𝑉, (𝐸𝐹)⟩ ∈ UMGraph )

Theoremusgredg2 40419 The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (#‘(𝐸𝑋)) = 2)

Theoremusgredg2ALT 40420 Alternate proof of usgredg2 40419, not using umgredg2 25766. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (#‘(𝐸𝑋)) = 2)

Theoremusgredgprv 40421 In a simple graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

TheoremusgredgprvALT 40422 Alternate proof of usgredgprv 40421, using usgredg2 40419 instead of umgredgprv 25773. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} → (𝑀𝑉𝑁𝑉)))

Theoremusgredgappr 40423 An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge). Analogue of usgredg2 40419. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → (#‘𝐶) = 2)

Theoremusgrpredgav 40424 An edge of a simple graph always connects two vertices. Analogue of usgredgprv 40421. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremedgassv2 40425 An edge of a simple graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → (𝐶𝑉 ∧ (#‘𝐶) = 2))

Theoremusgredg 40426* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Shortened by AV, 25-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝐶 = {𝑎, 𝑏}))

Theoremusgrnloopv 40427 In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

TheoremusgrnloopvALT 40428 Alternate proof of usgrnloopv 40427, not using umgrnloopv 25772. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀𝑊) → ((𝐸𝑋) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgrnloop 40429* In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

TheoremusgrnloopALT 40430* Alternate proof of usgrnloop 40429, not using umgrnloop 25774. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → (∃𝑥 ∈ dom 𝐸(𝐸𝑥) = {𝑀, 𝑁} → 𝑀𝑁))

Theoremusgrnloop0 40431* A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremusgrnloop0ALT 40432* Alternate proof of usgrnloop0 40431, not using umgrnloop0 25775. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) = {𝑈}} = ∅)

Theoremusgredgne 40433 An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv 40427 resp. usgrnloop 40429. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → 𝑀𝑁)

Theoremusgrf1oedg 40434 The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto𝐸)

Theoremuhgr2edg 40435* 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 40436* 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 40437* 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 40438* 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 40439* 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 40440* 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 40437. (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 40441* If a vertex is adjacent to two different vertices in a multigraph, there is not only one edge starting at this vertex, analogous to usgra2edg1 25912. 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 40442 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 40443* 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 40444* 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 40445* 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 40446* 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 40447* 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 40448* 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 40449* Alternate proof of usgredg2vtxeu 40448, using edgiedgb 25798, 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 40450* Lemma for uspgredg2v 40451. (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 40451* 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 40452* Lemma 1 for usgredg2v 40454. (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 40453* Lemma 2 for usgredg2v 40454. (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 40454* 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)

Theoremusgredgleord 40455* 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))

Theoremushgredgedga 40456* 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)

Theoremusgredgedga 40457* 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 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)

Theoremushgredgedgaloop 40458* In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex and the set of loops at this vertex. (Contributed by AV, 11-Dec-2020.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)    &   𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼𝑖) = {𝑁}}    &   𝐵 = {𝑒𝐸𝑒 = {𝑁}}    &   𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))       ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)

Theoremuspgredgaleord 40459* 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))

Theoremusgredgaleord 40460* 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))

TheoremusgredgaleordALT 40461* 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.) TODO-AV: proof can be shortened by using "bj-eleq2w", after it is moved to main.set.
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (#‘{𝑒𝐸𝑁𝑒}) ≤ (#‘𝑉))

21.34.8.3  Examples for graphs

Theoremusgr0e 40462 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 40463 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 40464 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 40465 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 40466 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 40467 The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph )

Theoremuhgr0vusgr 40468 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 40469 The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
∅ ∈ USGraph

Theoremuspgr1e 40470 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 40471 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 40472 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 40473 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 40474 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 40475 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 40476 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 40477 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 40478 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 40479 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 40480* 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 40481 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 40482 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 40483 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) = ∅))

Theoremusgrexmpllem 40484 Lemma for usgrexmpl 40487. (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 40485 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 40486 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 40487 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 40488* 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 40489* 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 40490 The class of simple graphs is a proper class (and therefore, because of prcssprc 40306, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.)
USGraph ∉ V

21.34.8.4  Subgraphs

Syntaxcsubgr 40491 Extend class notation with subgraphs.
class SubGraph

Definitiondf-subgr 40492* 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 40493 The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.)
Rel SubGraph

Theoremsubgrv 40494 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 40495 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 40496 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 40497 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 40498 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 40499 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 40500 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))

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