Theorem usgrf 29241

Description: The edge function of a simple graph is a one-to-one function into the set of proper unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)

Hypotheses

Ref	Expression
isuspgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isuspgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
usgrf	⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})

Distinct variable groups:   𝑥,𝐺   𝑥,𝑉

Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem usgrf

Step	Hyp	Ref	Expression
1		isuspgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		isuspgr.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	isusgr 29239	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
4	3	ibi 267	1 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3390   ∖ cdif 3887  ∅c0 4274  𝒫 cpw 4542  {csn 4568  dom cdm 5625  –1-1→wf1 6490  ‘cfv 6493  2c2 12230  ♯chash 14286  Vtxcvtx 29082  iEdgciedg 29083  USGraphcusgr 29235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fv 6501  df-usgr 29237

This theorem is referenced by:  usgredg2ALT  29279  usgrf1oedg  29293  usgrsizedg  29301  usgrres  29394