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Theorem uhgr0vb 25738
 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 2610 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2610 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2uhgrf 25728 . . 3 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 pweq 4111 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅)
54difeq1d 3689 . . . . . . 7 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅}))
6 pw0 4283 . . . . . . . . 9 𝒫 ∅ = {∅}
76difeq1i 3686 . . . . . . . 8 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
8 difid 3902 . . . . . . . 8 ({∅} ∖ {∅}) = ∅
97, 8eqtri 2632 . . . . . . 7 (𝒫 ∅ ∖ {∅}) = ∅
105, 9syl6eq 2660 . . . . . 6 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1110adantl 481 . . . . 5 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1211feq3d 5945 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
13 f00 6000 . . . . 5 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
1413simplbi 475 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
1512, 14syl6bi 242 . . 3 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺) = ∅))
163, 15syl5 33 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
17 simpl 472 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺𝑊)
18 simpr 476 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
1917, 18uhgr0e 25737 . . . 4 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph )
2019ex 449 . . 3 (𝐺𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph ))
2120adantr 480 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph ))
2216, 21impbid 201 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-uhgr 25724 This theorem is referenced by:  usgr0vb  40463  uhgr0v0e  40464  0uhgrsubgr  40503  0uhgrrusgr  40778  frgr0v  41433  frgruhgr0v  41435
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