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Theorem egrsubgr 28801
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 1135 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
2 eqid 2730 . . . . . . 7 (iEdg‘𝑆) = (iEdg‘𝑆)
3 eqid 2730 . . . . . . 7 (Edg‘𝑆) = (Edg‘𝑆)
42, 3edg0iedg0 28582 . . . . . 6 (Fun (iEdg‘𝑆) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
54adantl 480 . . . . 5 (((𝐺𝑊𝑆𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
6 res0 5984 . . . . . . 7 ((iEdg‘𝐺) ↾ ∅) = ∅
76eqcomi 2739 . . . . . 6 ∅ = ((iEdg‘𝐺) ↾ ∅)
8 id 22 . . . . . 6 ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ∅)
9 dmeq 5902 . . . . . . . 8 ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = dom ∅)
10 dm0 5919 . . . . . . . 8 dom ∅ = ∅
119, 10eqtrdi 2786 . . . . . . 7 ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = ∅)
1211reseq2d 5980 . . . . . 6 ((iEdg‘𝑆) = ∅ → ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) = ((iEdg‘𝐺) ↾ ∅))
137, 8, 123eqtr4a 2796 . . . . 5 ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
145, 13syl6bi 252 . . . 4 (((𝐺𝑊𝑆𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
1514impr 453 . . 3 (((𝐺𝑊𝑆𝑈) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
16153adant2 1129 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
17 0ss 4395 . . . . 5 ∅ ⊆ 𝒫 (Vtx‘𝑆)
18 sseq1 4006 . . . . 5 ((Edg‘𝑆) = ∅ → ((Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆) ↔ ∅ ⊆ 𝒫 (Vtx‘𝑆)))
1917, 18mpbiri 257 . . . 4 ((Edg‘𝑆) = ∅ → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
2019adantl 480 . . 3 ((Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
21203ad2ant3 1133 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
22 eqid 2730 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
23 eqid 2730 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
24 eqid 2730 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
2522, 23, 2, 24, 3issubgr 28795 . . 3 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
26253ad2ant1 1131 . 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 1340 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 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  wss 3947  c0 4321  𝒫 cpw 4601   class class class wbr 5147  dom cdm 5675  cres 5677  Fun wfun 6536  cfv 6542  Vtxcvtx 28523  iEdgciedg 28524  Edgcedg 28574   SubGraph csubgr 28791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fv 6550  df-edg 28575  df-subgr 28792
This theorem is referenced by:  0uhgrsubgr  28803
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