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Theorem uhgr0vb 29089
Description: The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.)
Assertion
Ref Expression
uhgr0vb ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))

Proof of Theorem uhgr0vb
StepHypRef Expression
1 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2uhgrf 29079 . . 3 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 pweq 4614 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅)
54difeq1d 4125 . . . . . . 7 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅}))
6 pw0 4812 . . . . . . . . 9 𝒫 ∅ = {∅}
76difeq1i 4122 . . . . . . . 8 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
8 difid 4376 . . . . . . . 8 ({∅} ∖ {∅}) = ∅
97, 8eqtri 2765 . . . . . . 7 (𝒫 ∅ ∖ {∅}) = ∅
105, 9eqtrdi 2793 . . . . . 6 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1110adantl 481 . . . . 5 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1211feq3d 6723 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
13 f00 6790 . . . . 5 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
1413simplbi 497 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
1512, 14biimtrdi 253 . . 3 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺) = ∅))
163, 15syl5 34 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
17 simpl 482 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺𝑊)
18 simpr 484 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
1917, 18uhgr0e 29088 . . . 4 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
2019ex 412 . . 3 (𝐺𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2120adantr 480 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2216, 21impbid 212 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  wf 6557  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-uhgr 29075
This theorem is referenced by:  usgr0vb  29254  uhgr0v0e  29255  0uhgrsubgr  29296  finsumvtxdg2size  29568  0uhgrrusgr  29596  frgr0v  30281  frgruhgr0v  30283
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