Theorem subgrprop2 16501

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.)

Hypotheses

Ref	Expression
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

Ref	Expression
subgrprop2	|- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ E C_ ~P V ) )

Proof of Theorem subgrprop2

Step	Hyp	Ref	Expression
1		issubgr.v	. . 3 |- V = (Vtx ` S )
2		issubgr.a	. . 3 |- A = (Vtx ` G )
3		issubgr.i	. . 3 |- I = (iEdg ` S )
4		issubgr.b	. . 3 |- B = (iEdg ` G )
5		issubgr.e	. . 3 |- E = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop 16500	. 2 |- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )
7		resss 5087	. . . 4 |- ( B |` dom I ) C_ B
8		sseq1 3271	. . . 4 |- ( I = ( B |` dom I ) -> ( I C_ B <-> ( B |` dom I ) C_ B ) )
9	7, 8	mpbiri 168	. . 3 |- ( I = ( B |` dom I ) -> I C_ B )
10	9	3anim2i 1217	. 2 |- ( ( 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 14	1 |- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ E C_ ~P V ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   C_ wss 3220   ~Pcpw 3688   class class class wbr 4130   dom cdm 4774   |` cres 4776   ` cfv 5377  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298   SubGraph csubgr 16494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-dm 4784  df-res 4786  df-iota 5337  df-fv 5385  df-subgr 16495

This theorem is used by:  uhgrissubgr  16502  subgrprop3  16503  subgrfun  16508  subgreldmiedg  16510  subgruhgredgdm  16511  subumgredg2en  16512  subuhgr  16513  subupgr  16514  subumgr  16515  subusgr  16516