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

Definition df-uspgr 25955
 Description: Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph (see uspgrupgr 25981) or a special undirected simple hypergraph (see uspgrushgr 25980), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") 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. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Assertion
Ref Expression
df-uspgr USPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-uspgr
StepHypRef Expression
1 cuspgr 25953 . 2 class USPGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1479 . . . . . . 7 class 𝑒
43cdm 5079 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1479 . . . . . . . . 9 class 𝑥
7 chash 13065 . . . . . . . . 9 class #
86, 7cfv 5852 . . . . . . . 8 class (#‘𝑥)
9 c2 11022 . . . . . . . 8 class 2
10 cle 10027 . . . . . . . 8 class
118, 9, 10wbr 4618 . . . . . . 7 wff (#‘𝑥) ≤ 2
12 vv . . . . . . . . . 10 setvar 𝑣
1312cv 1479 . . . . . . . . 9 class 𝑣
1413cpw 4135 . . . . . . . 8 class 𝒫 𝑣
15 c0 3896 . . . . . . . . 9 class
1615csn 4153 . . . . . . . 8 class {∅}
1714, 16cdif 3556 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1811, 5, 17crab 2911 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
194, 18, 3wf1 5849 . . . . 5 wff 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
20 vg . . . . . . 7 setvar 𝑔
2120cv 1479 . . . . . 6 class 𝑔
22 ciedg 25792 . . . . . 6 class iEdg
2321, 22cfv 5852 . . . . 5 class (iEdg‘𝑔)
2419, 2, 23wsbc 3421 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
25 cvtx 25791 . . . . 5 class Vtx
2621, 25cfv 5852 . . . 4 class (Vtx‘𝑔)
2724, 12, 26wsbc 3421 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
2827, 20cab 2607 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
291, 28wceq 1480 1 wff USPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
 Colors of variables: wff setvar class This definition is referenced by:  isuspgr  25957
 Copyright terms: Public domain W3C validator