Theorem uhgr0vb 29052

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	⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))

Proof of Theorem uhgr0vb

Step	Hyp	Ref	Expression
1		eqid 2733	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	uhgrf 29042	. . 3 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4		pweq 4563	. . . . . . . 8 ⊢ ((Vtx‘𝐺) = ∅ → 𝒫 (Vtx‘𝐺) = 𝒫 ∅)
5	4	difeq1d 4074	. . . . . . 7 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = (𝒫 ∅ ∖ {∅}))
6		pw0 4763	. . . . . . . . 9 ⊢ 𝒫 ∅ = {∅}
7	6	difeq1i 4071	. . . . . . . 8 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
8		difid 4325	. . . . . . . 8 ⊢ ({∅} ∖ {∅}) = ∅
9	7, 8	eqtri 2756	. . . . . . 7 ⊢ (𝒫 ∅ ∖ {∅}) = ∅
10	5, 9	eqtrdi 2784	. . . . . 6 ⊢ ((Vtx‘𝐺) = ∅ → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
11	10	adantl 481	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝒫 (Vtx‘𝐺) ∖ {∅}) = ∅)
12	11	feq3d 6641	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅))
13		f00 6710	. . . . 5 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ ↔ ((iEdg‘𝐺) = ∅ ∧ dom (iEdg‘𝐺) = ∅))
14	13	simplbi 497	. . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶∅ → (iEdg‘𝐺) = ∅)
15	12, 14	biimtrdi 253	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺) = ∅))
16	3, 15	syl5 34	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph → (iEdg‘𝐺) = ∅))
17		simpl 482	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊)
18		simpr 484	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅)
19	17, 18	uhgr0e 29051	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ UHGraph)
20	19	ex 412	. . 3 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
21	20	adantr 480	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ UHGraph))
22	16, 21	impbid 212	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3895  ∅c0 4282  𝒫 cpw 4549  {csn 4575  dom cdm 5619  ⟶wf 6482  ‘cfv 6486  Vtxcvtx 28976  iEdgciedg 28977  UHGraphcuhgr 29036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-uhgr 29038

This theorem is referenced by:  usgr0vb  29217  uhgr0v0e  29218  0uhgrsubgr  29259  finsumvtxdg2size  29531  0uhgrrusgr  29559  frgr0v  30244  frgruhgr0v  30246