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Theorem 0uhgrsubgr 29044
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 1145 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺𝑊𝑆 ∈ UHGraph))
2 0ss 4391 . . . 4 ∅ ⊆ (Vtx‘𝐺)
3 sseq1 4002 . . . 4 ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺)))
42, 3mpbiri 258 . . 3 ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
543ad2ant3 1132 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
6 eqid 2726 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
76uhgrfun 28834 . . 3 (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆))
873ad2ant2 1131 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆))
9 edgval 28817 . . 3 (Edg‘𝑆) = ran (iEdg‘𝑆)
10 uhgr0vb 28840 . . . . . . . 8 ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅))
11 rneq 5929 . . . . . . . . 9 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅)
12 rn0 5919 . . . . . . . . 9 ran ∅ = ∅
1311, 12eqtrdi 2782 . . . . . . . 8 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)
1410, 13biimtrdi 252 . . . . . . 7 ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅))
1514ex 412 . . . . . 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 1108 . . 3 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅)
199, 18eqtrid 2778 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅)
20 egrsubgr 29042 . 2 (((𝐺𝑊𝑆 ∈ UHGraph) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
211, 5, 8, 19, 20syl112anc 1371 1 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wss 3943  c0 4317   class class class wbr 5141  ran crn 5670  Fun wfun 6531  cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  UHGraphcuhgr 28824   SubGraph csubgr 29032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-edg 28816  df-uhgr 28826  df-subgr 29033
This theorem is referenced by: (None)
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