Theorem 0uhgrsubgr 16506

Description: The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.)

Assertion

Ref	Expression
0uhgrsubgr	|- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> S SubGraph G )

Proof of Theorem 0uhgrsubgr

Step	Hyp	Ref	Expression
1		3simpa 1025	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( G e. W /\ S e. UHGraph ) )
2		0ss 3561	. . . 4 |- (/) C_ (Vtx ` G )
3		sseq1 3271	. . . 4 |- ( (Vtx ` S ) = (/) -> ( (Vtx ` S ) C_ (Vtx ` G ) <-> (/) C_ (Vtx ` G ) ) )
4	2, 3	mpbiri 168	. . 3 |- ( (Vtx ` S ) = (/) -> (Vtx ` S ) C_ (Vtx ` G ) )
5	4	3ad2ant3 1051	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> (Vtx ` S ) C_ (Vtx ` G ) )
6		eqid 2238	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
7	6	uhgrfun 16318	. . 3 |- ( S e. UHGraph -> Fun (iEdg ` S ) )
8	7	3ad2ant2 1050	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> Fun (iEdg ` S ) )
9		edgval 16301	. . 3 |- (Edg ` S ) = ran (iEdg ` S )
10		uhgr0vb 16325	. . . . . . . 8 |- ( ( S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( S e. UHGraph <-> (iEdg ` S ) = (/) ) )
11		rneq 5009	. . . . . . . . 9 |- ( (iEdg ` S ) = (/) -> ran (iEdg ` S ) = ran (/) )
12		rn0 5038	. . . . . . . . 9 |- ran (/) = (/)
13	11, 12	eqtrdi 2287	. . . . . . . 8 |- ( (iEdg ` S ) = (/) -> ran (iEdg ` S ) = (/) )
14	10, 13	biimtrdi 163	. . . . . . 7 |- ( ( S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( S e. UHGraph -> ran (iEdg ` S ) = (/) ) )
15	14	ex 115	. . . . . 6 |- ( S e. UHGraph -> ( (Vtx ` S ) = (/) -> ( S e. UHGraph -> ran (iEdg ` S ) = (/) ) ) )
16	15	pm2.43a 51	. . . . 5 |- ( S e. UHGraph -> ( (Vtx ` S ) = (/) -> ran (iEdg ` S ) = (/) ) )
17	16	a1i 9	. . . 4 |- ( G e. W -> ( S e. UHGraph -> ( (Vtx ` S ) = (/) -> ran (iEdg ` S ) = (/) ) ) )
18	17	3imp 1224	. . 3 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ran (iEdg ` S ) = (/) )
19	9, 18	eqtrid 2283	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> (Edg ` S ) = (/) )
20		egrsubgr 16504	. 2 |- ( ( ( G e. W /\ S e. UHGraph ) /\ (Vtx ` S ) C_ (Vtx ` G ) /\ ( Fun (iEdg ` S ) /\ (Edg ` S ) = (/) ) ) -> S SubGraph G )
21	1, 5, 8, 19, 20	syl112anc 1282	1 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> S SubGraph G )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520   class class class wbr 4130   ran crn 4775   Fun wfun 5371   ` cfv 5377  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  UHGraphcuhgr 16308   SubGraph csubgr 16494

This proof depends on 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 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-subgr 16495

This theorem is used by: (None)