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Theorem 0uhgrsubgr 26216
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 1078 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺𝑊𝑆 ∈ UHGraph))
2 0ss 4005 . . . 4 ∅ ⊆ (Vtx‘𝐺)
3 sseq1 3659 . . . 4 ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺)))
42, 3mpbiri 248 . . 3 ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
543ad2ant3 1104 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
6 eqid 2651 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
76uhgrfun 26006 . . 3 (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆))
873ad2ant2 1103 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆))
9 edgval 25986 . . 3 (Edg‘𝑆) = ran (iEdg‘𝑆)
10 uhgr0vb 26012 . . . . . . . 8 ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅))
11 rneq 5383 . . . . . . . . 9 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅)
12 rn0 5409 . . . . . . . . 9 ran ∅ = ∅
1311, 12syl6eq 2701 . . . . . . . 8 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)
1410, 13syl6bi 243 . . . . . . 7 ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅))
1514ex 449 . . . . . 6 (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅)))
1615pm2.43a 54 . . . . 5 (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅))
1716a1i 11 . . . 4 (𝐺𝑊 → (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)))
18173imp 1275 . . 3 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅)
199, 18syl5eq 2697 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅)
20 egrsubgr 26214 . 2 (((𝐺𝑊𝑆 ∈ UHGraph) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
211, 5, 8, 19, 20syl112anc 1370 1 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  c0 3948   class class class wbr 4685  ran crn 5144  Fun wfun 5920  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UHGraphcuhgr 25996   SubGraph csubgr 26204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-edg 25985  df-uhgr 25998  df-subgr 26205
This theorem is referenced by: (None)
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