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Metamath Proof Explorer

Theorem subgrprop

Description: The properties of a subgraph. (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 subgrprop 
|- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ 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 subgrv 
 |-  ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
7 1 2 3 4 5 issubgr 
 |-  ( ( G e. _V /\ S e. _V ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
8 7 biimpd 
 |-  ( ( G e. _V /\ S e. _V ) -> ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
9 8 ancoms 
 |-  ( ( S e. _V /\ G e. _V ) -> ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
10 6 9 mpcom 
 |-  ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )