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Definition df-upgr 40289
Description: Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set 𝑣 (of "vertices") and a 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. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgr 40290). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.)
Assertion
Ref Expression
df-upgr UPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-upgr
StepHypRef Expression
1 cupgr 40287 . 2 class UPGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1473 . . . . . . 7 class 𝑒
43cdm 5027 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1473 . . . . . . . . 9 class 𝑥
7 chash 12936 . . . . . . . . 9 class #
86, 7cfv 5789 . . . . . . . 8 class (#‘𝑥)
9 c2 10919 . . . . . . . 8 class 2
10 cle 9931 . . . . . . . 8 class
118, 9, 10wbr 4577 . . . . . . 7 wff (#‘𝑥) ≤ 2
12 vv . . . . . . . . . 10 setvar 𝑣
1312cv 1473 . . . . . . . . 9 class 𝑣
1413cpw 4107 . . . . . . . 8 class 𝒫 𝑣
15 c0 3873 . . . . . . . . 9 class
1615csn 4124 . . . . . . . 8 class {∅}
1714, 16cdif 3536 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1811, 5, 17crab 2899 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
194, 18, 3wf 5785 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
20 vg . . . . . . 7 setvar 𝑔
2120cv 1473 . . . . . 6 class 𝑔
22 ciedg 40211 . . . . . 6 class iEdg
2321, 22cfv 5789 . . . . 5 class (iEdg‘𝑔)
2419, 2, 23wsbc 3401 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
25 cvtx 40210 . . . . 5 class Vtx
2621, 25cfv 5789 . . . 4 class (Vtx‘𝑔)
2724, 12, 26wsbc 3401 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
2827, 20cab 2595 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
291, 28wceq 1474 1 wff UPGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
Colors of variables: wff setvar class
This definition is referenced by:  isupgr  40291
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