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Theorem uhgr0e 29089
Description: The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)
Hypotheses
Ref Expression
uhgr0e.g (𝜑𝐺𝑊)
uhgr0e.e (𝜑 → (iEdg‘𝐺) = ∅)
Assertion
Ref Expression
uhgr0e (𝜑𝐺 ∈ UHGraph)

Proof of Theorem uhgr0e
StepHypRef Expression
1 f0 6788 . . 3 ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
2 dm0 5930 . . . 4 dom ∅ = ∅
32feq2i 6727 . . 3 (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
41, 3mpbir 231 . 2 ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
5 uhgr0e.g . . . 4 (𝜑𝐺𝑊)
6 eqid 2736 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2736 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isuhgr 29078 . . . 4 (𝐺𝑊 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
95, 8syl 17 . . 3 (𝜑 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
10 uhgr0e.e . . . 4 (𝜑 → (iEdg‘𝐺) = ∅)
11 id 22 . . . . 5 ((iEdg‘𝐺) = ∅ → (iEdg‘𝐺) = ∅)
12 dmeq 5913 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
1311, 12feq12d 6723 . . . 4 ((iEdg‘𝐺) = ∅ → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
1410, 13syl 17 . . 3 (𝜑 → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
159, 14bitrd 279 . 2 (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
164, 15mpbiri 258 1 (𝜑𝐺 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  cdif 3947  c0 4332  𝒫 cpw 4599  {csn 4625  dom cdm 5684  wf 6556  cfv 6560  Vtxcvtx 29014  iEdgciedg 29015  UHGraphcuhgr 29074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-uhgr 29076
This theorem is referenced by:  uhgr0vb  29090
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