Theorem egrsubgr 16113

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 1024	. 2 |- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> (Vtx ` S ) C_ (Vtx ` G ) )
2		eqid 2231	. . . . . . 7 |- (iEdg ` S ) = (iEdg ` S )
3		eqid 2231	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
4	2, 3	edg0iedg0g 15916	. . . . . 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 5017	. . . . . . 7 |- ( (iEdg ` G ) |` (/) ) = (/)
7	6	eqcomi 2235	. . . . . 6 |- (/) = ( (iEdg ` G ) |` (/) )
8		id 19	. . . . . 6 |- ( (iEdg ` S ) = (/) -> (iEdg ` S ) = (/) )
9		dmeq 4931	. . . . . . . 8 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = dom (/) )
10		dm0 4945	. . . . . . . 8 |- dom (/) = (/)
11	9, 10	eqtrdi 2280	. . . . . . 7 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = (/) )
12	11	reseq2d 5013	. . . . . 6 |- ( (iEdg ` S ) = (/) -> ( (iEdg ` G ) |` dom (iEdg ` S ) ) = ( (iEdg ` G ) |` (/) ) )
13	7, 8, 12	3eqtr4a 2290	. . . . 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 1042	. 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 3533	. . . . 5 |- (/) C_ ~P (Vtx ` S )
18		sseq1 3250	. . . . 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 1046	. 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 2231	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
23		eqid 2231	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
24		eqid 2231	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
25	22, 23, 2, 24, 3	issubgr 16107	. . 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 1044	. 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 1206	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 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   (/)c0 3494   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   |` cres 4727   Fun wfun 5320   ` 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-in1 619  ax-in2 620  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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  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-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-edgf 15855  df-iedg 15865  df-edg 15908  df-subgr 16104

This theorem is referenced by:  0uhgrsubgr  16115