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Theorem egrsubgr 27365
Description: An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 17-Dec-2020.)
Assertion
Ref Expression
egrsubgr (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)

Proof of Theorem egrsubgr
StepHypRef Expression
1 simp2 1139 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
2 eqid 2737 . . . . . . 7 (iEdg‘𝑆) = (iEdg‘𝑆)
3 eqid 2737 . . . . . . 7 (Edg‘𝑆) = (Edg‘𝑆)
42, 3edg0iedg0 27146 . . . . . 6 (Fun (iEdg‘𝑆) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
54adantl 485 . . . . 5 (((𝐺𝑊𝑆𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
6 res0 5855 . . . . . . 7 ((iEdg‘𝐺) ↾ ∅) = ∅
76eqcomi 2746 . . . . . 6 ∅ = ((iEdg‘𝐺) ↾ ∅)
8 id 22 . . . . . 6 ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ∅)
9 dmeq 5772 . . . . . . . 8 ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = dom ∅)
10 dm0 5789 . . . . . . . 8 dom ∅ = ∅
119, 10eqtrdi 2794 . . . . . . 7 ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = ∅)
1211reseq2d 5851 . . . . . 6 ((iEdg‘𝑆) = ∅ → ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) = ((iEdg‘𝐺) ↾ ∅))
137, 8, 123eqtr4a 2804 . . . . 5 ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
145, 13syl6bi 256 . . . 4 (((𝐺𝑊𝑆𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
1514impr 458 . . 3 (((𝐺𝑊𝑆𝑈) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
16153adant2 1133 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
17 0ss 4311 . . . . 5 ∅ ⊆ 𝒫 (Vtx‘𝑆)
18 sseq1 3926 . . . . 5 ((Edg‘𝑆) = ∅ → ((Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆) ↔ ∅ ⊆ 𝒫 (Vtx‘𝑆)))
1917, 18mpbiri 261 . . . 4 ((Edg‘𝑆) = ∅ → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
2019adantl 485 . . 3 ((Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
21203ad2ant3 1137 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
22 eqid 2737 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
23 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
24 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
2522, 23, 2, 24, 3issubgr 27359 . . 3 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
26253ad2ant1 1135 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
271, 16, 21, 26mpbir3and 1344 1 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wss 3866  c0 4237  𝒫 cpw 4513   class class class wbr 5053  dom cdm 5551  cres 5553  Fun wfun 6374  cfv 6380  Vtxcvtx 27087  iEdgciedg 27088  Edgcedg 27138   SubGraph csubgr 27355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-iota 6338  df-fun 6382  df-fv 6388  df-edg 27139  df-subgr 27356
This theorem is referenced by:  0uhgrsubgr  27367
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