Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-umgra Structured version   Visualization version   GIF version

Definition df-umgra 25842
 Description: Define the class of all undirected multigraphs. A multigraph is a pair ⟨𝑉, 𝐸⟩ where 𝐸 is a function into subsets of 𝑉 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
df-umgra UMGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
Distinct variable group:   𝑣,𝑒,𝑥

Detailed syntax breakdown of Definition df-umgra
StepHypRef Expression
1 cumg 25841 . 2 class UMGrph
2 ve . . . . . 6 setvar 𝑒
32cv 1474 . . . . 5 class 𝑒
43cdm 5038 . . . 4 class dom 𝑒
5 vx . . . . . . . 8 setvar 𝑥
65cv 1474 . . . . . . 7 class 𝑥
7 chash 12979 . . . . . . 7 class #
86, 7cfv 5804 . . . . . 6 class (#‘𝑥)
9 c2 10947 . . . . . 6 class 2
10 cle 9954 . . . . . 6 class
118, 9, 10wbr 4583 . . . . 5 wff (#‘𝑥) ≤ 2
12 vv . . . . . . . 8 setvar 𝑣
1312cv 1474 . . . . . . 7 class 𝑣
1413cpw 4108 . . . . . 6 class 𝒫 𝑣
15 c0 3874 . . . . . . 7 class
1615csn 4125 . . . . . 6 class {∅}
1714, 16cdif 3537 . . . . 5 class (𝒫 𝑣 ∖ {∅})
1811, 5, 17crab 2900 . . . 4 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
194, 18, 3wf 5800 . . 3 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
2019, 12, 2copab 4642 . 2 class {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
211, 20wceq 1475 1 wff UMGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
 Colors of variables: wff setvar class This definition is referenced by:  relumgra  25843  isumgra  25844
 Copyright terms: Public domain W3C validator