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Definition df-usgr 26267
Description: Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr 26294), consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Assertion
Ref Expression
df-usgr USGraph = {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-usgr
StepHypRef Expression
1 cusgr 26265 . 2 class USGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1630 . . . . . . 7 class 𝑒
43cdm 5250 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1630 . . . . . . . . 9 class 𝑥
7 chash 13320 . . . . . . . . 9 class
86, 7cfv 6030 . . . . . . . 8 class (♯‘𝑥)
9 c2 11275 . . . . . . . 8 class 2
108, 9wceq 1631 . . . . . . 7 wff (♯‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1630 . . . . . . . . 9 class 𝑣
1312cpw 4298 . . . . . . . 8 class 𝒫 𝑣
14 c0 4063 . . . . . . . . 9 class
1514csn 4317 . . . . . . . 8 class {∅}
1613, 15cdif 3720 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 3065 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
184, 17, 3wf1 6027 . . . . 5 wff 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1630 . . . . . 6 class 𝑔
21 ciedg 26095 . . . . . 6 class iEdg
2220, 21cfv 6030 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3587 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24 cvtx 26094 . . . . 5 class Vtx
2520, 24cfv 6030 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3587 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
2726, 19cab 2757 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
281, 27wceq 1631 1 wff USGraph = {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:  isusgr  26269
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