Theorem 0uhgrsubgr 26171

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 1058	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( G e. W /\ S e. UHGraph ) )
2		0ss 3972	. . . 4 |- (/) C_ (Vtx ` G )
3		sseq1 3626	. . . 4 |- ( (Vtx ` S ) = (/) -> ( (Vtx ` S ) C_ (Vtx ` G ) <-> (/) C_ (Vtx ` G ) ) )
4	2, 3	mpbiri 248	. . 3 |- ( (Vtx ` S ) = (/) -> (Vtx ` S ) C_ (Vtx ` G ) )
5	4	3ad2ant3 1084	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> (Vtx ` S ) C_ (Vtx ` G ) )
6		eqid 2622	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
7	6	uhgrfun 25961	. . 3 |- ( S e. UHGraph -> Fun (iEdg ` S ) )
8	7	3ad2ant2 1083	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> Fun (iEdg ` S ) )
9		edgval 25941	. . 3 |- (Edg ` S ) = ran (iEdg ` S )
10		uhgr0vb 25967	. . . . . . . 8 |- ( ( S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( S e. UHGraph <-> (iEdg ` S ) = (/) ) )
11		rneq 5351	. . . . . . . . 9 |- ( (iEdg ` S ) = (/) -> ran (iEdg ` S ) = ran (/) )
12		rn0 5377	. . . . . . . . 9 |- ran (/) = (/)
13	11, 12	syl6eq 2672	. . . . . . . 8 |- ( (iEdg ` S ) = (/) -> ran (iEdg ` S ) = (/) )
14	10, 13	syl6bi 243	. . . . . . 7 |- ( ( S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( S e. UHGraph -> ran (iEdg ` S ) = (/) ) )
15	14	ex 450	. . . . . 6 |- ( S e. UHGraph -> ( (Vtx ` S ) = (/) -> ( S e. UHGraph -> ran (iEdg ` S ) = (/) ) ) )
16	15	pm2.43a 54	. . . . 5 |- ( S e. UHGraph -> ( (Vtx ` S ) = (/) -> ran (iEdg ` S ) = (/) ) )
17	16	a1i 11	. . . 4 |- ( G e. W -> ( S e. UHGraph -> ( (Vtx ` S ) = (/) -> ran (iEdg ` S ) = (/) ) ) )
18	17	3imp 1256	. . 3 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ran (iEdg ` S ) = (/) )
19	9, 18	syl5eq 2668	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> (Edg ` S ) = (/) )
20		egrsubgr 26169	. 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 1330	1 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> S SubGraph G )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ran crn 5115   Fun wfun 5882   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   SubGraph csubgr 26159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-edg 25940  df-uhgr 25953  df-subgr 26160

This theorem is referenced by: (None)