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Theorem 0grsubgr 26361
Description: The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)
Assertion
Ref Expression
0grsubgr (𝐺𝑊 → ∅ SubGraph 𝐺)

Proof of Theorem 0grsubgr
StepHypRef Expression
1 0ss 4107 . . 3 ∅ ⊆ (Vtx‘𝐺)
2 dm0 5486 . . . . 5 dom ∅ = ∅
32reseq2i 5540 . . . 4 ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4 res0 5547 . . . 4 ((iEdg‘𝐺) ↾ ∅) = ∅
53, 4eqtr2i 2775 . . 3 ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6 0ss 4107 . . 3 ∅ ⊆ 𝒫 ∅
71, 5, 63pm3.2i 1421 . 2 (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8 0ex 4934 . . 3 ∅ ∈ V
9 vtxval0 26122 . . . . 5 (Vtx‘∅) = ∅
109eqcomi 2761 . . . 4 ∅ = (Vtx‘∅)
11 eqid 2752 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
12 iedgval0 26123 . . . . 5 (iEdg‘∅) = ∅
1312eqcomi 2761 . . . 4 ∅ = (iEdg‘∅)
14 eqid 2752 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
15 edgval 26132 . . . . 5 (Edg‘∅) = ran (iEdg‘∅)
1612rneqi 5499 . . . . 5 ran (iEdg‘∅) = ran ∅
17 rn0 5524 . . . . 5 ran ∅ = ∅
1815, 16, 173eqtrri 2779 . . . 4 ∅ = (Edg‘∅)
1910, 11, 13, 14, 18issubgr 26354 . . 3 ((𝐺𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
208, 19mpan2 709 . 2 (𝐺𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
217, 20mpbiri 248 1 (𝐺𝑊 → ∅ SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1624  wcel 2131  Vcvv 3332  wss 3707  c0 4050  𝒫 cpw 4294   class class class wbr 4796  dom cdm 5258  ran crn 5259  cres 5260  cfv 6041  Vtxcvtx 26065  iEdgciedg 26066  Edgcedg 26130   SubGraph csubgr 26350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-iota 6004  df-fun 6043  df-fv 6049  df-slot 16055  df-base 16057  df-edgf 26059  df-vtx 26067  df-iedg 26068  df-edg 26131  df-subgr 26351
This theorem is referenced by: (None)
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