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Theorem upgredgss 29164
Description: The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020.)
Assertion
Ref Expression
upgredgss (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
Distinct variable group:   𝑥,𝐺

Proof of Theorem upgredgss
StepHypRef Expression
1 edgval 29081 . 2 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 eqid 2735 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2735 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3upgrf 29118 . . 3 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
54frnd 6745 . 2 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
61, 5eqsstrid 4044 1 (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  {crab 3433  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  dom cdm 5689  ran crn 5690  cfv 6563  cle 11294  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  UPGraphcupgr 29112
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  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-mpt 5232  df-id 5583  df-xp 5695  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-fv 6571  df-edg 29080  df-upgr 29114
This theorem is referenced by:  uspgrupgrushgr  29211  upgredgssspr  47987
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