Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  upgredgss Structured version   Visualization version   GIF version

Theorem upgredgss 26909
 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 26826 . 2 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 eqid 2819 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2819 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
42, 3upgrf 26863 . . 3 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
54frnd 6514 . 2 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
61, 5eqsstrid 4013 1 (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2108  {crab 3140   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537  {csn 4559   class class class wbr 5057  dom cdm 5548  ran crn 5549  ‘cfv 6348   ≤ cle 10668  2c2 11684  ♯chash 13682  Vtxcvtx 26773  iEdgciedg 26774  Edgcedg 26824  UPGraphcupgr 26857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-edg 26825  df-upgr 26859 This theorem is referenced by:  uspgrupgrushgr  26954  upgredgssspr  44009
 Copyright terms: Public domain W3C validator