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Theorem usgrf 29187
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 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem usgrf
StepHypRef Expression
1 isuspgr.v . . 3 𝑉 = (Vtx‘𝐺)
2 isuspgr.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2isusgr 29185 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}))
43ibi 267 1 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  cdif 3960  c0 4339  𝒫 cpw 4605  {csn 4631  dom cdm 5689  1-1wf1 6560  cfv 6563  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  USGraphcusgr 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571  df-usgr 29183
This theorem is referenced by:  usgredg2ALT  29225  usgrf1oedg  29239  usgrsizedg  29247  usgrres  29340
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