Theorem subgrprop 16109

Description: The properties of a subgraph. (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
subgrprop	|- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )

Proof of Theorem subgrprop

Step	Hyp	Ref	Expression
1		subgrv 16106	. 2 |- ( S SubGraph G -> ( S e. _V /\ G e. _V ) )
2		issubgr.v	. . . . 5 |- V = (Vtx ` S )
3		issubgr.a	. . . . 5 |- A = (Vtx ` G )
4		issubgr.i	. . . . 5 |- I = (iEdg ` S )
5		issubgr.b	. . . . 5 |- B = (iEdg ` G )
6		issubgr.e	. . . . 5 |- E = (Edg ` S )
7	2, 3, 4, 5, 6	issubgr 16107	. . . 4 |- ( ( G e. _V /\ S e. _V ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
8	7	biimpd 144	. . 3 |- ( ( G e. _V /\ S e. _V ) -> ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
9	8	ancoms 268	. 2 |- ( ( S e. _V /\ G e. _V ) -> ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) ) )
10	1, 9	mpcom 36	1 |- ( S SubGraph G -> ( V C_ A /\ I = ( B |` dom I ) /\ E C_ ~P V ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   |` cres 4727   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907   SubGraph csubgr 16103

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-dm 4735  df-res 4737  df-iota 5286  df-fv 5334  df-subgr 16104

This theorem is referenced by:  subgrprop2  16110