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Theorem egrsubgr 27077
 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 1134 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
2 eqid 2798 . . . . . . 7 (iEdg‘𝑆) = (iEdg‘𝑆)
3 eqid 2798 . . . . . . 7 (Edg‘𝑆) = (Edg‘𝑆)
42, 3edg0iedg0 26858 . . . . . 6 (Fun (iEdg‘𝑆) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
54adantl 485 . . . . 5 (((𝐺𝑊𝑆𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
6 res0 5823 . . . . . . 7 ((iEdg‘𝐺) ↾ ∅) = ∅
76eqcomi 2807 . . . . . 6 ∅ = ((iEdg‘𝐺) ↾ ∅)
8 id 22 . . . . . 6 ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ∅)
9 dmeq 5737 . . . . . . . 8 ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = dom ∅)
10 dm0 5755 . . . . . . . 8 dom ∅ = ∅
119, 10eqtrdi 2849 . . . . . . 7 ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = ∅)
1211reseq2d 5819 . . . . . 6 ((iEdg‘𝑆) = ∅ → ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) = ((iEdg‘𝐺) ↾ ∅))
137, 8, 123eqtr4a 2859 . . . . 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 1128 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
17 0ss 4304 . . . . 5 ∅ ⊆ 𝒫 (Vtx‘𝑆)
18 sseq1 3940 . . . . 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 1132 . 2 (((𝐺𝑊𝑆𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
22 eqid 2798 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
23 eqid 2798 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
24 eqid 2798 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
2522, 23, 2, 24, 3issubgr 27071 . . 3 ((𝐺𝑊𝑆𝑈) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
26253ad2ant1 1130 . 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 1339 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 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5031  dom cdm 5520   ↾ cres 5522  Fun wfun 6319  ‘cfv 6325  Vtxcvtx 26799  iEdgciedg 26800  Edgcedg 26850   SubGraph csubgr 27067 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-iota 6284  df-fun 6327  df-fv 6333  df-edg 26851  df-subgr 27068 This theorem is referenced by:  0uhgrsubgr  27079
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