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Theorem uhgr0vb 27442
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 2738 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2738 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2uhgrf 27432 . . 3 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 pweq 4549 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅)
54difeq1d 4056 . . . . . . 7 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅}))
6 pw0 4745 . . . . . . . . 9 𝒫 ∅ = {∅}
76difeq1i 4053 . . . . . . . 8 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
8 difid 4304 . . . . . . . 8 ({∅} ∖ {∅}) = ∅
97, 8eqtri 2766 . . . . . . 7 (𝒫 ∅ ∖ {∅}) = ∅
105, 9eqtrdi 2794 . . . . . 6 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1110adantl 482 . . . . 5 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1211feq3d 6587 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
13 f00 6656 . . . . 5 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
1413simplbi 498 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
1512, 14syl6bi 252 . . 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 483 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺𝑊)
18 simpr 485 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
1917, 18uhgr0e 27441 . . . 4 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
2019ex 413 . . 3 (𝐺𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2120adantr 481 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2216, 21impbid 211 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  cdif 3884  c0 4256  𝒫 cpw 4533  {csn 4561  dom cdm 5589  wf 6429  cfv 6433  Vtxcvtx 27366  iEdgciedg 27367  UHGraphcuhgr 27426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-uhgr 27428
This theorem is referenced by:  usgr0vb  27604  uhgr0v0e  27605  0uhgrsubgr  27646  finsumvtxdg2size  27917  0uhgrrusgr  27945  frgr0v  28626  frgruhgr0v  28628
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