Theorem usgrf 29089

Description: The edge function of a simple graph is a one-to-one function into 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 29087	. 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 1540   ∈ wcel 2109  {crab 3408   ∖ cdif 3914  ∅c0 4299  𝒫 cpw 4566  {csn 4592  dom cdm 5641  –1-1→wf1 6511  ‘cfv 6514  2c2 12248  ♯chash 14302  Vtxcvtx 28930  iEdgciedg 28931  USGraphcusgr 29083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fv 6522  df-usgr 29085

This theorem is referenced by:  usgredg2ALT  29127  usgrf1oedg  29141  usgrsizedg  29149  usgrres  29242