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Database SUPPLEMENTARY MATERIAL (USERS' MATHBOXES) Mathbox for Alexander van der Vekens Graph theory (extension) Isomorphisms of graphs grimid
Description: The identity relation restricted to the set of vertices of a graph is a
graph isomorphism between the graph and itself. (Contributed by AV , 29-Apr-2025) (Prove shortened by AV, 5-May-2025.)
Ref
Expression
Assertion
grimid ⊢ ( 𝐺 ∈ UHGraph → ( I ↾ ( Vtx ‘ 𝐺 ) ) ∈ ( 𝐺 GraphIso 𝐺 ) )
Proof
Step
Hyp
Ref
Expression
1
id ⊢ ( 𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph )
2
eqidd ⊢ ( 𝐺 ∈ UHGraph → ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) )
3
eqidd ⊢ ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) )
4
1 1 2 3
grimidvtxedg ⊢ ( 𝐺 ∈ UHGraph → ( I ↾ ( Vtx ‘ 𝐺 ) ) ∈ ( 𝐺 GraphIso 𝐺 ) )