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Theorem uspgrf 25942
Description: The edge function of a simple pseudograph 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
uspgrf (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem uspgrf
StepHypRef Expression
1 isuspgr.v . . 3 𝑉 = (Vtx‘𝐺)
2 isuspgr.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2isuspgr 25940 . 2 (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
43ibi 256 1 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2911  cdif 3552  c0 3891  𝒫 cpw 4130  {csn 4148   class class class wbr 4613  dom cdm 5074  1-1wf1 5844  cfv 5847  cle 10019  2c2 11014  #chash 13057  Vtxcvtx 25774  iEdgciedg 25775   USPGraph cuspgr 25936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fv 5855  df-uspgr 25938
This theorem is referenced by:  uspgrf1oedg  25961  usgrumgruspgr  25968  usgruspgrb  25969  usgrislfuspgr  25972  uspgrn2crct  26569
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