Metamath Proof Explorer


Theorem usgrexmpl

Description: 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) (Proof shortened by AV, 7-Aug-2025)

Ref Expression
Hypotheses usgrexmpl.v V = 0 4
usgrexmpl.e E = ⟨“ 0 1 1 2 2 0 0 3 ”⟩
usgrexmpl.g G = V E
Assertion usgrexmpl G USGraph

Proof

Step Hyp Ref Expression
1 usgrexmpl.v V = 0 4
2 usgrexmpl.e E = ⟨“ 0 1 1 2 2 0 0 3 ”⟩
3 usgrexmpl.g G = V E
4 1 2 usgrexmplef E : dom E 1-1 e 𝒫 V | e = 2
5 3 eleq1i G USGraph V E USGraph
6 1 ovexi V V
7 s4cli ⟨“ 0 1 1 2 2 0 0 3 ”⟩ Word V
8 2 7 eqeltri E Word V
9 isusgrop V V E Word V V E USGraph E : dom E 1-1 e 𝒫 V | e = 2
10 6 8 9 mp2an V E USGraph E : dom E 1-1 e 𝒫 V | e = 2
11 5 10 bitri G USGraph E : dom E 1-1 e 𝒫 V | e = 2
12 4 11 mpbir G USGraph