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Theorem 0grsubgr 26575
 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 4197 . . 3 ∅ ⊆ (Vtx‘𝐺)
2 dm0 5571 . . . . 5 dom ∅ = ∅
32reseq2i 5626 . . . 4 ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4 res0 5633 . . . 4 ((iEdg‘𝐺) ↾ ∅) = ∅
53, 4eqtr2i 2850 . . 3 ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6 0ss 4197 . . 3 ∅ ⊆ 𝒫 ∅
71, 5, 63pm3.2i 1442 . 2 (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8 0ex 5014 . . 3 ∅ ∈ V
9 vtxval0 26337 . . . . 5 (Vtx‘∅) = ∅
109eqcomi 2834 . . . 4 ∅ = (Vtx‘∅)
11 eqid 2825 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
12 iedgval0 26338 . . . . 5 (iEdg‘∅) = ∅
1312eqcomi 2834 . . . 4 ∅ = (iEdg‘∅)
14 eqid 2825 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
15 edgval 26347 . . . . 5 (Edg‘∅) = ran (iEdg‘∅)
1612rneqi 5584 . . . . 5 ran (iEdg‘∅) = ran ∅
17 rn0 5610 . . . . 5 ran ∅ = ∅
1815, 16, 173eqtrri 2854 . . . 4 ∅ = (Edg‘∅)
1910, 11, 13, 14, 18issubgr 26568 . . 3 ((𝐺𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
208, 19mpan2 682 . 2 (𝐺𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
217, 20mpbiri 250 1 (𝐺𝑊 → ∅ SubGraph 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  𝒫 cpw 4378   class class class wbr 4873  dom cdm 5342  ran crn 5343   ↾ cres 5344  ‘cfv 6123  Vtxcvtx 26294  iEdgciedg 26295  Edgcedg 26345   SubGraph csubgr 26564 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fv 6131  df-slot 16226  df-base 16228  df-edgf 26288  df-vtx 26296  df-iedg 26297  df-edg 26346  df-subgr 26565 This theorem is referenced by: (None)
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