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Theorem uhgr0e 26783
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 6553 . . 3 ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
2 dm0 5783 . . . 4 dom ∅ = ∅
32feq2i 6499 . . 3 (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
41, 3mpbir 232 . 2 ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})
5 uhgr0e.g . . . 4 (𝜑𝐺𝑊)
6 eqid 2818 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
7 eqid 2818 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7isuhgr 26772 . . . 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 5765 . . . . 5 ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
1311, 12feq12d 6495 . . . 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 280 . 2 (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
164, 15mpbiri 259 1 (𝜑𝐺 ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  cdif 3930  c0 4288  𝒫 cpw 4535  {csn 4557  dom cdm 5548  wf 6344  cfv 6348  Vtxcvtx 26708  iEdgciedg 26709  UHGraphcuhgr 26768
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-pr 5320
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-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-uhgr 26770
This theorem is referenced by:  uhgr0vb  26784
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