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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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