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Theorem upgrf 26787
 Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 26788 without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Hypotheses
Ref Expression
isupgr.v 𝑉 = (Vtx‘𝐺)
isupgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
upgrf (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem upgrf
StepHypRef Expression
1 isupgr.v . . 3 𝑉 = (Vtx‘𝐺)
2 isupgr.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2isupgr 26785 . 2 (𝐺 ∈ UPGraph → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
43ibi 268 1 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2107  {crab 3146   ∖ cdif 3936  ∅c0 4294  𝒫 cpw 4541  {csn 4563   class class class wbr 5062  dom cdm 5553  ⟶wf 6347  ‘cfv 6351   ≤ cle 10668  2c2 11684  ♯chash 13683  Vtxcvtx 26697  iEdgciedg 26698  UPGraphcupgr 26781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-upgr 26783 This theorem is referenced by:  upgrfn  26788  upgrss  26789  upgrop  26795  upgruhgr  26803  upgrun  26819  umgrislfupgr  26824  upgredgss  26833  edgupgr  26835  upgredg  26838  upgrreslem  27002  upgrres1  27011
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