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Theorem uhgr0vb 26166
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 2760 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2760 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2uhgrf 26156 . . 3 (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 pweq 4305 . . . . . . . 8 ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅)
54difeq1d 3870 . . . . . . 7 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅}))
6 pw0 4488 . . . . . . . . 9 𝒫 ∅ = {∅}
76difeq1i 3867 . . . . . . . 8 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
8 difid 4091 . . . . . . . 8 ({∅} ∖ {∅}) = ∅
97, 8eqtri 2782 . . . . . . 7 (𝒫 ∅ ∖ {∅}) = ∅
105, 9syl6eq 2810 . . . . . 6 ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1110adantl 473 . . . . 5 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
1211feq3d 6193 . . . 4 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
13 f00 6248 . . . . 5 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
1413simplbi 478 . . . 4 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
1512, 14syl6bi 243 . . 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 474 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺𝑊)
18 simpr 479 . . . . 5 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
1917, 18uhgr0e 26165 . . . 4 ((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
2019ex 449 . . 3 (𝐺𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2120adantr 472 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
2216, 21impbid 202 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  cdif 3712  c0 4058  𝒫 cpw 4302  {csn 4321  dom cdm 5266  wf 6045  cfv 6049  Vtxcvtx 26073  iEdgciedg 26074  UHGraphcuhgr 26150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-uhgr 26152
This theorem is referenced by:  usgr0vb  26328  uhgr0v0e  26329  0uhgrsubgr  26370  finsumvtxdg2size  26656  0uhgrrusgr  26684  frgr0v  27415  frgruhgr0v  27417
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