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Theorem uhgrsubgrself 28114
Description: A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
Assertion
Ref Expression
uhgrsubgrself (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

Proof of Theorem uhgrsubgrself
StepHypRef Expression
1 ssid 3964 . . 3 (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2 ssid 3964 . . 3 (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
31, 2pm3.2i 471 . 2 ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4 eqid 2736 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
54uhgrfun 27903 . . 3 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6 id 22 . . 3 (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7 eqid 2736 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
87, 7, 4, 4uhgrissubgr 28109 . . 3 ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
95, 6, 8mpd3an23 1463 . 2 (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
103, 9mpbiri 257 1 (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wss 3908   class class class wbr 5103  Fun wfun 6487  cfv 6493  Vtxcvtx 27833  iEdgciedg 27834  UHGraphcuhgr 27893   SubGraph csubgr 28101
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-edg 27885  df-uhgr 27895  df-subgr 28102
This theorem is referenced by: (None)
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