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Theorem 0uhgrsubgr 26058
 Description: The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.)
Assertion
Ref Expression
0uhgrsubgr ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)

Proof of Theorem 0uhgrsubgr
StepHypRef Expression
1 3simpa 1056 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺𝑊𝑆 ∈ UHGraph ))
2 0ss 3949 . . . 4 ∅ ⊆ (Vtx‘𝐺)
3 sseq1 3610 . . . 4 ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺)))
42, 3mpbiri 248 . . 3 ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
543ad2ant3 1082 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
6 eqid 2626 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
76uhgrfun 25852 . . 3 (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆))
873ad2ant2 1081 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆))
9 edgval 25836 . . . 4 (𝑆 ∈ UHGraph → (Edg‘𝑆) = ran (iEdg‘𝑆))
1093ad2ant2 1081 . . 3 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ran (iEdg‘𝑆))
11 uhgr0vb 25858 . . . . . . . 8 ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅))
12 rneq 5315 . . . . . . . . 9 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅)
13 rn0 5341 . . . . . . . . 9 ran ∅ = ∅
1412, 13syl6eq 2676 . . . . . . . 8 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)
1511, 14syl6bi 243 . . . . . . 7 ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅))
1615ex 450 . . . . . 6 (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅)))
1716pm2.43a 54 . . . . 5 (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅))
1817a1i 11 . . . 4 (𝐺𝑊 → (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)))
19183imp 1254 . . 3 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅)
2010, 19eqtrd 2660 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅)
21 egrsubgr 26056 . 2 (((𝐺𝑊𝑆 ∈ UHGraph ) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
221, 5, 8, 20, 21syl112anc 1327 1 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992   ⊆ wss 3560  ∅c0 3896   class class class wbr 4618  ran crn 5080  Fun wfun 5844  ‘cfv 5850  Vtxcvtx 25769  iEdgciedg 25770  Edgcedg 25834   UHGraph cuhgr 25842   SubGraph csubgr 26046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-edg 25835  df-uhgr 25844  df-subgr 26047 This theorem is referenced by: (None)
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