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Theorem 0grsubgr 26063
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 3944 . . 3 ∅ ⊆ (Vtx‘𝐺)
2 dm0 5299 . . . . 5 dom ∅ = ∅
32reseq2i 5353 . . . 4 ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4 res0 5360 . . . 4 ((iEdg‘𝐺) ↾ ∅) = ∅
53, 4eqtr2i 2644 . . 3 ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6 0ss 3944 . . 3 ∅ ⊆ 𝒫 ∅
71, 5, 63pm3.2i 1237 . 2 (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8 0ex 4750 . . 3 ∅ ∈ V
9 vtxval0 25831 . . . . 5 (Vtx‘∅) = ∅
109eqcomi 2630 . . . 4 ∅ = (Vtx‘∅)
11 eqid 2621 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
12 iedgval0 25832 . . . . 5 (iEdg‘∅) = ∅
1312eqcomi 2630 . . . 4 ∅ = (iEdg‘∅)
14 eqid 2621 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
15 edgval 25841 . . . . . 6 (∅ ∈ V → (Edg‘∅) = ran (iEdg‘∅))
168, 15ax-mp 5 . . . . 5 (Edg‘∅) = ran (iEdg‘∅)
1712rneqi 5312 . . . . 5 ran (iEdg‘∅) = ran ∅
18 rn0 5337 . . . . 5 ran ∅ = ∅
1916, 17, 183eqtrri 2648 . . . 4 ∅ = (Edg‘∅)
2010, 11, 13, 14, 19issubgr 26056 . . 3 ((𝐺𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
218, 20mpan2 706 . 2 (𝐺𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
227, 21mpbiri 248 1 (𝐺𝑊 → ∅ SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   class class class wbr 4613  dom cdm 5074  ran crn 5075  cres 5076  cfv 5847  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   SubGraph csubgr 26052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-slot 15785  df-base 15786  df-edgf 25768  df-vtx 25776  df-iedg 25777  df-edg 25840  df-subgr 26053
This theorem is referenced by: (None)
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