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Theorem usgra0v 25900
 Description: The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
Assertion
Ref Expression
usgra0v (∅ USGrph 𝐸𝐸 = ∅)

Proof of Theorem usgra0v
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . . 3 (∅ USGrph 𝐸 → (∅ ∈ V ∧ 𝐸 ∈ V))
2 isusgra 25873 . . . 4 ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2}))
3 eqidd 2611 . . . . . . 7 (𝐸 ∈ V → 𝐸 = 𝐸)
4 eqidd 2611 . . . . . . 7 (𝐸 ∈ V → dom 𝐸 = dom 𝐸)
5 pw0 4283 . . . . . . . . . . . 12 𝒫 ∅ = {∅}
65difeq1i 3686 . . . . . . . . . . 11 (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
7 difid 3902 . . . . . . . . . . 11 ({∅} ∖ {∅}) = ∅
86, 7eqtri 2632 . . . . . . . . . 10 (𝒫 ∅ ∖ {∅}) = ∅
98a1i 11 . . . . . . . . 9 (𝐸 ∈ V → (𝒫 ∅ ∖ {∅}) = ∅)
109rabeqdv 3167 . . . . . . . 8 (𝐸 ∈ V → {𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ ∅ ∣ (#‘𝑥) = 2})
11 rab0 3909 . . . . . . . 8 {𝑥 ∈ ∅ ∣ (#‘𝑥) = 2} = ∅
1210, 11syl6eq 2660 . . . . . . 7 (𝐸 ∈ V → {𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} = ∅)
133, 4, 12f1eq123d 6044 . . . . . 6 (𝐸 ∈ V → (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝐸1-1→∅))
14 f1f 6014 . . . . . . 7 (𝐸:dom 𝐸1-1→∅ → 𝐸:dom 𝐸⟶∅)
15 f00 6000 . . . . . . . 8 (𝐸:dom 𝐸⟶∅ ↔ (𝐸 = ∅ ∧ dom 𝐸 = ∅))
1615simplbi 475 . . . . . . 7 (𝐸:dom 𝐸⟶∅ → 𝐸 = ∅)
1714, 16syl 17 . . . . . 6 (𝐸:dom 𝐸1-1→∅ → 𝐸 = ∅)
1813, 17syl6bi 242 . . . . 5 (𝐸 ∈ V → (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸 = ∅))
1918adantl 481 . . . 4 ((∅ ∈ V ∧ 𝐸 ∈ V) → (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸 = ∅))
202, 19sylbid 229 . . 3 ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ USGrph 𝐸𝐸 = ∅))
211, 20mpcom 37 . 2 (∅ USGrph 𝐸𝐸 = ∅)
22 0ex 4718 . . . 4 ∅ ∈ V
23 usgra0 25899 . . . 4 (∅ ∈ V → ∅ USGrph ∅)
2422, 23ax-mp 5 . . 3 ∅ USGrph ∅
25 breq2 4587 . . 3 (𝐸 = ∅ → (∅ USGrph 𝐸 ↔ ∅ USGrph ∅))
2624, 25mpbiri 247 . 2 (𝐸 = ∅ → ∅ USGrph 𝐸)
2721, 26impbii 198 1 (∅ USGrph 𝐸𝐸 = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859 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-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-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-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-usgra 25862 This theorem is referenced by:  usgra1v  25919  usgrafisindb0  25937  frgra0v  26520
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