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Theorem uhgrsubgrself 26989
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 3986 . . 3 (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2 ssid 3986 . . 3 (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
31, 2pm3.2i 471 . 2 ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4 eqid 2818 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
54uhgrfun 26778 . . 3 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6 id 22 . . 3 (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7 eqid 2818 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
87, 7, 4, 4uhgrissubgr 26984 . . 3 ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
95, 6, 8mpd3an23 1454 . 2 (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
103, 9mpbiri 259 1 (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wss 3933   class class class wbr 5057  Fun wfun 6342  cfv 6348  Vtxcvtx 26708  iEdgciedg 26709  UHGraphcuhgr 26768   SubGraph csubgr 26976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-edg 26760  df-uhgr 26770  df-subgr 26977
This theorem is referenced by: (None)
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