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 𝐸 ↔ 𝐸 = ∅)

Proof of Theorem usgra0v

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrav 25867	. . 3 ⊢ (∅ USGrph 𝐸 → (∅ ∈ V ∧ 𝐸 ∈ V))
2		isusgra 25873	. . . 4 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ USGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{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 ∅ = {∅}
6	5	difeq1i 3686	. . . . . . . . . . 11 ⊢ (𝒫 ∅ ∖ {∅}) = ({∅} ∖ {∅})
7		difid 3902	. . . . . . . . . . 11 ⊢ ({∅} ∖ {∅}) = ∅
8	6, 7	eqtri 2632	. . . . . . . . . 10 ⊢ (𝒫 ∅ ∖ {∅}) = ∅
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝐸 ∈ V → (𝒫 ∅ ∖ {∅}) = ∅)
10	9	rabeqdv 3167	. . . . . . . 8 ⊢ (𝐸 ∈ V → {𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ ∅ ∣ (#‘𝑥) = 2})
11		rab0 3909	. . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ (#‘𝑥) = 2} = ∅
12	10, 11	syl6eq 2660	. . . . . . 7 ⊢ (𝐸 ∈ V → {𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} = ∅)
13	3, 4, 12	f1eq123d 6044	. . . . . 6 ⊢ (𝐸 ∈ V → (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→∅))
14		f1f 6014	. . . . . . 7 ⊢ (𝐸:dom 𝐸–1-1→∅ → 𝐸:dom 𝐸⟶∅)
15		f00 6000	. . . . . . . 8 ⊢ (𝐸:dom 𝐸⟶∅ ↔ (𝐸 = ∅ ∧ dom 𝐸 = ∅))
16	15	simplbi 475	. . . . . . 7 ⊢ (𝐸:dom 𝐸⟶∅ → 𝐸 = ∅)
17	14, 16	syl 17	. . . . . 6 ⊢ (𝐸:dom 𝐸–1-1→∅ → 𝐸 = ∅)
18	13, 17	syl6bi 242	. . . . 5 ⊢ (𝐸 ∈ V → (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸 = ∅))
19	18	adantl 481	. . . 4 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ V) → (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸 = ∅))
20	2, 19	sylbid 229	. . 3 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ USGrph 𝐸 → 𝐸 = ∅))
21	1, 20	mpcom 37	. 2 ⊢ (∅ USGrph 𝐸 → 𝐸 = ∅)
22		0ex 4718	. . . 4 ⊢ ∅ ∈ V
23		usgra0 25899	. . . 4 ⊢ (∅ ∈ V → ∅ USGrph ∅)
24	22, 23	ax-mp 5	. . 3 ⊢ ∅ USGrph ∅
25		breq2 4587	. . 3 ⊢ (𝐸 = ∅ → (∅ USGrph 𝐸 ↔ ∅ USGrph ∅))
26	24, 25	mpbiri 247	. 2 ⊢ (𝐸 = ∅ → ∅ USGrph 𝐸)
27	21, 26	impbii 198	1 ⊢ (∅ USGrph 𝐸 ↔ 𝐸 = ∅)

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