Definition df-subgr

Description: Define the class of the subgraph relation. A class s is a subgraph of a class g (thesupergraph of s ) if its vertices are also vertices of g , and its edges are also edges of g , connecting vertices of s only (see section I.1 in Bollobas p. 2 or section 1.1 in Diestel p. 4). The second condition is ensured by the requirement that the edge function of s is a restriction of the edge function of g having only vertices of s in its range. Note that the domains of the edge functions of the subgraph and the supergraph should be compatible. (Contributed by AV, 16-Nov-2020)

	Ref	Expression
Assertion	df-subgr	|- SubGraph = { <. s , g >. | ( ( Vtx ` s ) C_ ( Vtx ` g ) /\ ( iEdg ` s ) = ( ( iEdg ` g ) |` dom ( iEdg ` s ) ) /\ ( Edg ` s ) C_ ~P ( Vtx ` s ) ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		csubgr	|- SubGraph
1		vs	|- s
2		vg	|- g
3		cvtx	|- Vtx
4	1	cv	|- s
5	4 3	cfv	|- ( Vtx ` s )
6	2	cv	|- g
7	6 3	cfv	|- ( Vtx ` g )
8	5 7	wss	|- ( Vtx ` s ) C_ ( Vtx ` g )
9		ciedg	|- iEdg
10	4 9	cfv	|- ( iEdg ` s )
11	6 9	cfv	|- ( iEdg ` g )
12	10	cdm	|- dom ( iEdg ` s )
13	11 12	cres	|- ( ( iEdg ` g ) |` dom ( iEdg ` s ) )
14	10 13	wceq	|- ( iEdg ` s ) = ( ( iEdg ` g ) |` dom ( iEdg ` s ) )
15		cedg	|- Edg
16	4 15	cfv	|- ( Edg ` s )
17	5	cpw	|- ~P ( Vtx ` s )
18	16 17	wss	|- ( Edg ` s ) C_ ~P ( Vtx ` s )
19	8 14 18	w3a	|- ( ( Vtx ` s ) C_ ( Vtx ` g ) /\ ( iEdg ` s ) = ( ( iEdg ` g ) |` dom ( iEdg ` s ) ) /\ ( Edg ` s ) C_ ~P ( Vtx ` s ) )
20	19 1 2	copab	|- { <. s , g >. | ( ( Vtx ` s ) C_ ( Vtx ` g ) /\ ( iEdg ` s ) = ( ( iEdg ` g ) |` dom ( iEdg ` s ) ) /\ ( Edg ` s ) C_ ~P ( Vtx ` s ) ) }
21	0 20	wceq	|- SubGraph = { <. s , g >. | ( ( Vtx ` s ) C_ ( Vtx ` g ) /\ ( iEdg ` s ) = ( ( iEdg ` g ) |` dom ( iEdg ` s ) ) /\ ( Edg ` s ) C_ ~P ( Vtx ` s ) ) }