Theorem egrsubgr 16258

Description: An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 17-Dec-2020.)

Assertion

Ref	Expression
egrsubgr	|- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> S SubGraph G )

Proof of Theorem egrsubgr

Step	Hyp	Ref	Expression
1		simp2 1025	. 2 |- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> (Vtx ` S ) C_ (Vtx ` G ) )
2		eqid 2232	. . . . . . 7 |- (iEdg ` S ) = (iEdg ` S )
3		eqid 2232	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
4	2, 3	edg0iedg0g 16061	. . . . . 6 |- ( ( S e. U /\ Fun (iEdg ` S ) ) -> ( (Edg ` S ) = (/) <-> (iEdg ` S ) = (/) ) )
5	4	adantll 476	. . . . 5 |- ( ( ( G e. W /\ S e. U ) /\ Fun (iEdg ` S ) ) -> ( (Edg ` S ) = (/) <-> (iEdg ` S ) = (/) ) )
6		res0 5042	. . . . . . 7 |- ( (iEdg ` G ) |` (/) ) = (/)
7	6	eqcomi 2236	. . . . . 6 |- (/) = ( (iEdg ` G ) |` (/) )
8		id 19	. . . . . 6 |- ( (iEdg ` S ) = (/) -> (iEdg ` S ) = (/) )
9		dmeq 4956	. . . . . . . 8 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = dom (/) )
10		dm0 4970	. . . . . . . 8 |- dom (/) = (/)
11	9, 10	eqtrdi 2281	. . . . . . 7 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = (/) )
12	11	reseq2d 5038	. . . . . 6 |- ( (iEdg ` S ) = (/) -> ( (iEdg ` G ) |` dom (iEdg ` S ) ) = ( (iEdg ` G ) |` (/) ) )
13	7, 8, 12	3eqtr4a 2291	. . . . 5 |- ( (iEdg ` S ) = (/) -> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
14	5, 13	biimtrdi 163	. . . 4 |- ( ( ( G e. W /\ S e. U ) /\ Fun (iEdg ` S ) ) -> ( (Edg ` S ) = (/) -> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) ) )
15	14	impr 379	. . 3 |- ( ( ( G e. W /\ S e. U ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
16	15	3adant2 1043	. 2 |- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) )
17		0ss 3547	. . . . 5 |- (/) C_ ~P (Vtx ` S )
18		sseq1 3261	. . . . 5 |- ( (Edg ` S ) = (/) -> ( (Edg ` S ) C_ ~P (Vtx ` S ) <-> (/) C_ ~P (Vtx ` S ) ) )
19	17, 18	mpbiri 168	. . . 4 |- ( (Edg ` S ) = (/) -> (Edg ` S ) C_ ~P (Vtx ` S ) )
20	19	adantl 277	. . 3 |- ( ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) -> (Edg ` S ) C_ ~P (Vtx ` S ) )
21	20	3ad2ant3 1047	. 2 |- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> (Edg ` S ) C_ ~P (Vtx ` S ) )
22		eqid 2232	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
23		eqid 2232	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
24		eqid 2232	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
25	22, 23, 2, 24, 3	issubgr 16252	. . 3 |- ( ( G e. W /\ S e. U ) -> ( S SubGraph G <-> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) ) )
26	25	3ad2ant1 1045	. 2 |- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> ( S SubGraph G <-> ( (Vtx ` S ) C_ (Vtx ` G ) /\ (iEdg ` S ) = ( (iEdg ` G ) |` dom (iEdg ` S ) ) /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) ) )
27	1, 16, 21, 26	mpbir3and 1207	1 |- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> S SubGraph G )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   C_ wss 3211   (/)c0 3508   ~Pcpw 3669   class class class wbr 4109   dom cdm 4749   |` cres 4751   Fun wfun 5346   ` 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-in1 619  ax-in2 620  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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-2nd 6335  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-edgf 16000  df-iedg 16010  df-edg 16053  df-subgr 16249

This theorem is referenced by:  0uhgrsubgr  16260