Theorem 0uhgrsubgr 16115

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 1020	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( G e. W /\ S e. UHGraph ) )
2		0ss 3533	. . . 4 |- (/) C_ (Vtx ` G )
3		sseq1 3250	. . . 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 1046	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> (Vtx ` S ) C_ (Vtx ` G ) )
6		eqid 2231	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
7	6	uhgrfun 15927	. . 3 |- ( S e. UHGraph -> Fun (iEdg ` S ) )
8	7	3ad2ant2 1045	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> Fun (iEdg ` S ) )
9		edgval 15910	. . 3 |- (Edg ` S ) = ran (iEdg ` S )
10		uhgr0vb 15934	. . . . . . . 8 |- ( ( S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ( S e. UHGraph <-> (iEdg ` S ) = (/) ) )
11		rneq 4959	. . . . . . . . 9 |- ( (iEdg ` S ) = (/) -> ran (iEdg ` S ) = ran (/) )
12		rn0 4988	. . . . . . . . 9 |- ran (/) = (/)
13	11, 12	eqtrdi 2280	. . . . . . . 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 1219	. . 3 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> ran (iEdg ` S ) = (/) )
19	9, 18	eqtrid 2276	. 2 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> (Edg ` S ) = (/) )
20		egrsubgr 16113	. 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 1277	1 |- ( ( G e. W /\ S e. UHGraph /\ (Vtx ` S ) = (/) ) -> S SubGraph G )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   (/)c0 3494   class class class wbr 4088   ran crn 4726   Fun wfun 5320   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UHGraphcuhgr 15917   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-nul 4215  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-ima 4738  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-1st 6302  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-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-uhgrm 15919  df-subgr 16104

This theorem is referenced by: (None)