Definition df-upgr

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 ). (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 24-Nov-2020)

	Ref	Expression
Assertion	df-upgr	$$⊢\mathrm{UPGraph}=\left\{g|\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶\left\{x\in \left(𝒫v\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}\right\}$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cupgr	$$class\mathrm{UPGraph}$$
1		vg	$$setvarg$$
2		cvtx	$$class\mathrm{Vtx}$$
3	1	cv	$$setvarg$$
4	3 2	cfv	$$class\mathrm{Vtx}\left(g\right)$$
5		vv	$$setvarv$$
6		ciedg	$$class\mathrm{iEdg}$$
7	3 6	cfv	$$class\mathrm{iEdg}\left(g\right)$$
8		ve	$$setvare$$
9	8	cv	$$setvare$$
10	9	cdm	$$class\mathrm{dom}e$$
11		vx	$$setvarx$$
12	5	cv	$$setvarv$$
13	12	cpw	$$class𝒫v$$
14		c0	$$class\varnothing$$
15	14	csn	$$class\left\{\varnothing \right\}$$
16	13 15	cdif	$$class\left(𝒫v\setminus \left\{\varnothing \right\}\right)$$
17		chash	$$class\left|.\right|$$
18	11	cv	$$setvarx$$
19	18 17	cfv	$$class\left|x\right|$$
20		cle	$$class\le$$
21		c2	$$class2$$
22	19 21 20	wbr	$$wff\left|x\right|\le 2$$
23	22 11 16	crab	$$class\left\{x\in \left(𝒫v\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$
24	10 23 9	wf	$$wffe:\mathrm{dom}e⟶\left\{x\in \left(𝒫v\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$
25	24 8 7	wsbc	$$wff\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶\left\{x\in \left(𝒫v\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$
26	25 5 4	wsbc	$$wff\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶\left\{x\in \left(𝒫v\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$
27	26 1	cab	$$class\left\{g|\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶\left\{x\in \left(𝒫v\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}\right\}$$
28	0 27	wceq	$$wff\mathrm{UPGraph}=\left\{g|\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶\left\{x\in \left(𝒫v\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}\right\}$$