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Theorem 0grsubgr 27066
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 4322 . . 3 ∅ ⊆ (Vtx‘𝐺)
2 dm0 5767 . . . . 5 dom ∅ = ∅
32reseq2i 5828 . . . 4 ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4 res0 5835 . . . 4 ((iEdg‘𝐺) ↾ ∅) = ∅
53, 4eqtr2i 2846 . . 3 ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6 0ss 4322 . . 3 ∅ ⊆ 𝒫 ∅
71, 5, 63pm3.2i 1336 . 2 (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8 0ex 5187 . . 3 ∅ ∈ V
9 vtxval0 26830 . . . . 5 (Vtx‘∅) = ∅
109eqcomi 2831 . . . 4 ∅ = (Vtx‘∅)
11 eqid 2822 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
12 iedgval0 26831 . . . . 5 (iEdg‘∅) = ∅
1312eqcomi 2831 . . . 4 ∅ = (iEdg‘∅)
14 eqid 2822 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
15 edgval 26840 . . . . 5 (Edg‘∅) = ran (iEdg‘∅)
1612rneqi 5784 . . . . 5 ran (iEdg‘∅) = ran ∅
17 rn0 5773 . . . . 5 ran ∅ = ∅
1815, 16, 173eqtrri 2850 . . . 4 ∅ = (Edg‘∅)
1910, 11, 13, 14, 18issubgr 27059 . . 3 ((𝐺𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
208, 19mpan2 690 . 2 (𝐺𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
217, 20mpbiri 261 1 (𝐺𝑊 → ∅ SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2114  Vcvv 3469  wss 3908  c0 4265  𝒫 cpw 4511   class class class wbr 5042  dom cdm 5532  ran crn 5533  cres 5534  cfv 6334  Vtxcvtx 26787  iEdgciedg 26788  Edgcedg 26838   SubGraph csubgr 27055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fv 6342  df-slot 16478  df-base 16480  df-edgf 26781  df-vtx 26789  df-iedg 26790  df-edg 26839  df-subgr 27056
This theorem is referenced by: (None)
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