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Theorem uhgr0e 25862
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 6043 . . 3 ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
2 dm0 5299 . . . 4 dom ∅ = ∅
32feq2i 5994 . . 3 (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
41, 3mpbir 221 . 2 ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
5 uhgr0e.g . . . 4 (𝜑𝐺𝑊)
6 eqid 2621 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2621 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isuhgr 25851 . . . 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 5284 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
1311, 12feq12d 5990 . . . 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 268 . 2 (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
164, 15mpbiri 248 1 (𝜑𝐺 ∈ UHGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  cdif 3552  c0 3891  𝒫 cpw 4130  {csn 4148  dom cdm 5074  wf 5843  cfv 5847  Vtxcvtx 25774  iEdgciedg 25775   UHGraph cuhgr 25847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-uhgr 25849
This theorem is referenced by:  uhgr0vb  25863
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