Theorem usgraf 25875

Description: The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)

Assertion

Ref	Expression
usgraf	⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})

Distinct variable groups:   𝑥,𝐸   𝑥,𝑉

Proof of Theorem usgraf

Step	Hyp	Ref	Expression
1		usgrav 25867	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2		isusgra 25873	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
3	1, 2	syl 17	. 2 ⊢ (𝑉 USGrph 𝐸 → (𝑉 USGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))
4	3	ibi 255	1 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})

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  –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-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:  usgrares  25898  usgraedg2  25904  constr3trllem1  26178  constr3trllem5  26182