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Theorem uhgrsubgrself 29312
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 4018 . . 3 (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2 ssid 4018 . . 3 (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
31, 2pm3.2i 470 . 2 ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4 eqid 2735 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
54uhgrfun 29098 . . 3 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6 id 22 . . 3 (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7 eqid 2735 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
87, 7, 4, 4uhgrissubgr 29307 . . 3 ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
95, 6, 8mpd3an23 1462 . 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 2106  wss 3963   class class class wbr 5148  Fun wfun 6557  cfv 6563  Vtxcvtx 29028  iEdgciedg 29029  UHGraphcuhgr 29088   SubGraph csubgr 29299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-edg 29080  df-uhgr 29090  df-subgr 29300
This theorem is referenced by: (None)
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