Theorem egrsubgr 16418

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 1029	. 2 |- ( ( ( G e. W /\ S e. U ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> (Vtx ` S ) C_ (Vtx ` G ) )
2		eqid 2238	. . . . . . 7 |- (iEdg ` S ) = (iEdg ` S )
3		eqid 2238	. . . . . . 7 |- (Edg ` S ) = (Edg ` S )
4	2, 3	edg0iedg0g 16221	. . . . . 6 |- ( ( S e. U /\ Fun (iEdg ` S ) ) -> ( (Edg ` S ) = (/) <-> (iEdg ` S ) = (/) ) )
5	4	adantll 480	. . . . 5 |- ( ( ( G e. W /\ S e. U ) /\ Fun (iEdg ` S ) ) -> ( (Edg ` S ) = (/) <-> (iEdg ` S ) = (/) ) )
6		res0 5062	. . . . . . 7 |- ( (iEdg ` G ) |` (/) ) = (/)
7	6	eqcomi 2242	. . . . . 6 |- (/) = ( (iEdg ` G ) |` (/) )
8		id 19	. . . . . 6 |- ( (iEdg ` S ) = (/) -> (iEdg ` S ) = (/) )
9		dmeq 4976	. . . . . . . 8 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = dom (/) )
10		dm0 4990	. . . . . . . 8 |- dom (/) = (/)
11	9, 10	eqtrdi 2287	. . . . . . 7 |- ( (iEdg ` S ) = (/) -> dom (iEdg ` S ) = (/) )
12	11	reseq2d 5058	. . . . . 6 |- ( (iEdg ` S ) = (/) -> ( (iEdg ` G ) |` dom (iEdg ` S ) ) = ( (iEdg ` G ) |` (/) ) )
13	7, 8, 12	3eqtr4a 2297	. . . . 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 1047	. 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 3561	. . . . 5 |- (/) C_ ~P (Vtx ` S )
18		sseq1 3271	. . . . 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 1051	. 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 2238	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
23		eqid 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
24		eqid 2238	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
25	22, 23, 2, 24, 3	issubgr 16412	. . 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 1049	. 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 1211	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 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   |` cres 4771   Fun wfun 5366   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212   SubGraph csubgr 16408

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-edgf 16160  df-iedg 16170  df-edg 16213  df-subgr 16409

This theorem is referenced by:  0uhgrsubgr  16420