Theorem subgrprop2

Description: The properties of a subgraph: If S is a subgraph of G , its vertices are also vertices of G , and its edges are also edges of G , connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020)

	Ref	Expression
Hypotheses	issubgr.v	|- V = ( Vtx ` S )
	issubgr.a	|- A = ( Vtx ` G )
	issubgr.i	|- I = ( iEdg ` S )
	issubgr.b	|- B = ( iEdg ` G )
	issubgr.e	|- E = ( Edg ` S )
Assertion	subgrprop2	|- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ E C_ ~P V ) )

Proof

Step	Hyp	Ref	Expression
1		issubgr.v	|- V = ( Vtx ` S )
2		issubgr.a	|- A = ( Vtx ` G )
3		issubgr.i	|- I = ( iEdg ` S )
4		issubgr.b	|- B = ( iEdg ` G )
5		issubgr.e	|- E = ( Edg ` S )
6	1 2 3 4 5	subgrprop	|- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )
7		resss	|- ( B |` dom I ) C_ B
8		sseq1	|- ( I = ( B |` dom I ) -> ( I C_ B <-> ( B |` dom I ) C_ B ) )
9	7 8	mpbiri	|- ( I = ( B |` dom I ) -> I C_ B )
10	9	3anim2i	|- ( ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) -> ( V C_ A /\ I C_ B /\ E C_ ~P V ) )
11	6 10	syl	|- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ E C_ ~P V ) )