Theorem subgrprop2 16255

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 16254	. 2 |- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )
7		resss 5062	. . . 4 |- ( B |` dom I ) C_ B
8		sseq1 3261	. . . 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 1213	. 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 ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   C_ wss 3211   ~Pcpw 3669   class class class wbr 4109   dom cdm 4749   |` cres 4751   ` cfv 5352  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052   SubGraph csubgr 16248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-dm 4759  df-res 4761  df-iota 5312  df-fv 5360  df-subgr 16249

This theorem is referenced by:  uhgrissubgr  16256  subgrprop3  16257  subgrfun  16262  subgreldmiedg  16264  subgruhgredgdm  16265  subumgredg2en  16266  subuhgr  16267  subupgr  16268  subumgr  16269  subusgr  16270