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Theorem uhgr0e 26086
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 6199 . . 3 ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
2 dm0 5446 . . . 4 dom ∅ = ∅
32feq2i 6150 . . 3 (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
41, 3mpbir 221 . 2 ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
5 uhgr0e.g . . . 4 (𝜑𝐺𝑊)
6 eqid 2724 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2724 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isuhgr 26075 . . . 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 5431 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
1311, 12feq12d 6146 . . . 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 1596  wcel 2103  cdif 3677  c0 4023  𝒫 cpw 4266  {csn 4285  dom cdm 5218  wf 5997  cfv 6001  Vtxcvtx 25994  iEdgciedg 25995  UHGraphcuhgr 26071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-uhgr 26073
This theorem is referenced by:  uhgr0vb  26087
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