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

Step	Hyp	Ref	Expression
1		cumg 25841	. 2 class UMGrph
2		ve	. . . . . 6 setvar 𝑒
3	2	cv 1474	. . . . 5 class 𝑒
4	3	cdm 5038	. . . 4 class dom 𝑒
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1474	. . . . . . 7 class 𝑥
7		chash 12979	. . . . . . 7 class #
8	6, 7	cfv 5804	. . . . . 6 class (#‘𝑥)
9		c2 10947	. . . . . 6 class 2
10		cle 9954	. . . . . 6 class ≤
11	8, 9, 10	wbr 4583	. . . . 5 wff (#‘𝑥) ≤ 2
12		vv	. . . . . . . 8 setvar 𝑣
13	12	cv 1474	. . . . . . 7 class 𝑣
14	13	cpw 4108	. . . . . 6 class 𝒫 𝑣
15		c0 3874	. . . . . . 7 class ∅
16	15	csn 4125	. . . . . 6 class {∅}
17	14, 16	cdif 3537	. . . . 5 class (𝒫 𝑣 ∖ {∅})
18	11, 5, 17	crab 2900	. . . 4 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
19	4, 18, 3	wf 5800	. . 3 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
20	19, 12, 2	copab 4642	. 2 class {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
21	1, 20	wceq 1475	1 wff UMGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

This definition is referenced by:  relumgra  25843  isumgra  25844