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Theorem 0uhgrsubgr 29352
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 1148 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺𝑊𝑆 ∈ UHGraph))
2 0ss 4352 . . . 4 ∅ ⊆ (Vtx‘𝐺)
3 sseq1 3959 . . . 4 ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺)))
42, 3mpbiri 258 . . 3 ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
543ad2ant3 1135 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
6 eqid 2736 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
76uhgrfun 29139 . . 3 (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆))
873ad2ant2 1134 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆))
9 edgval 29122 . . 3 (Edg‘𝑆) = ran (iEdg‘𝑆)
10 uhgr0vb 29145 . . . . . . . 8 ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅))
11 rneq 5885 . . . . . . . . 9 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅)
12 rn0 5875 . . . . . . . . 9 ran ∅ = ∅
1311, 12eqtrdi 2787 . . . . . . . 8 ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)
1410, 13biimtrdi 253 . . . . . . 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 1110 . . 3 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅)
199, 18eqtrid 2783 . 2 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅)
20 egrsubgr 29350 . 2 (((𝐺𝑊𝑆 ∈ UHGraph) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
211, 5, 8, 19, 20syl112anc 1376 1 ((𝐺𝑊𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1541  wcel 2113  wss 3901  c0 4285   class class class wbr 5098  ran crn 5625  Fun wfun 6486  cfv 6492  Vtxcvtx 29069  iEdgciedg 29070  Edgcedg 29120  UHGraphcuhgr 29129   SubGraph csubgr 29340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-edg 29121  df-uhgr 29131  df-subgr 29341
This theorem is referenced by: (None)
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