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Theorem uhgrsubgrself 29297
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 4006 . . 3 (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2 ssid 4006 . . 3 (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
31, 2pm3.2i 470 . 2 ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
54uhgrfun 29083 . . 3 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6 id 22 . . 3 (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
87, 7, 4, 4uhgrissubgr 29292 . . 3 ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
95, 6, 8mpd3an23 1465 . 2 (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
103, 9mpbiri 258 1 (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wss 3951   class class class wbr 5143  Fun wfun 6555  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073   SubGraph csubgr 29284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-edg 29065  df-uhgr 29075  df-subgr 29285
This theorem is referenced by: (None)
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