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Theorem uhgr0vb 26871
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 2824 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2824 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2uhgrf 26861 . . 3 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 pweq 4539 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅)
54difeq1d 4085 . . . . . . 7 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅}))
6 pw0 4730 . . . . . . . . 9 𝒫 ∅ = {∅}
76difeq1i 4082 . . . . . . . 8 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
8 difid 4314 . . . . . . . 8 ({∅} ∖ {∅}) = ∅
97, 8eqtri 2847 . . . . . . 7 (𝒫 ∅ ∖ {∅}) = ∅
105, 9syl6eq 2875 . . . . . 6 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1110adantl 485 . . . . 5 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1211feq3d 6491 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
13 f00 6552 . . . . 5 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
1413simplbi 501 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
1512, 14syl6bi 256 . . 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 486 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺𝑊)
18 simpr 488 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
1917, 18uhgr0e 26870 . . . 4 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
2019ex 416 . . 3 (𝐺𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2120adantr 484 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2216, 21impbid 215 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cdif 3917  c0 4277  𝒫 cpw 4523  {csn 4551  dom cdm 5543  wf 6340  cfv 6344  Vtxcvtx 26795  iEdgciedg 26796  UHGraphcuhgr 26855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-uhgr 26857
This theorem is referenced by:  usgr0vb  27033  uhgr0v0e  27034  0uhgrsubgr  27075  finsumvtxdg2size  27346  0uhgrrusgr  27374  frgr0v  28053  frgruhgr0v  28055
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