Theorem egrsubgr 16384

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 2234	. . . . . . 7 |- (iEdg ` S ) = (iEdg ` S )
3		eqid 2234	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
4	2, 3	edg0iedg0g 16187	. . . . . 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 5047	. . . . . . 7 |- ( (iEdg ` G ) |` (/) ) = (/)
7	6	eqcomi 2238	. . . . . 6 |- (/) = ( (iEdg ` G ) |` (/) )
8		id 19	. . . . . 6 |- ( (iEdg ` S ) = (/) -> (iEdg ` S ) = (/) )
9		dmeq 4961	. . . . . . . 8 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = dom (/) )
10		dm0 4975	. . . . . . . 8 |- dom (/) = (/)
11	9, 10	eqtrdi 2283	. . . . . . 7 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = (/) )
12	11	reseq2d 5043	. . . . . 6 |- ( (iEdg ` S ) = (/) -> ( (iEdg ` G ) |` dom (iEdg ` S ) ) = ( (iEdg ` G ) |` (/) ) )
13	7, 8, 12	3eqtr4a 2293	. . . . 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 3551	. . . . 5 |- (/) C_ ~P (Vtx ` S )
18		sseq1 3265	. . . . 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 2234	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
23		eqid 2234	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
24		eqid 2234	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
25	22, 23, 2, 24, 3	issubgr 16378	. . 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 2205   C_ wss 3214   (/)c0 3512   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   |` cres 4756   Fun wfun 5351   ` cfv 5357  Vtxcvtx 16133  iEdgciedg 16134  Edgcedg 16178   SubGraph csubgr 16374

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-2nd 6348  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-edgf 16126  df-iedg 16136  df-edg 16179  df-subgr 16375

This theorem is referenced by:  0uhgrsubgr  16386