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Theorem upgrf 26321
Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 26322 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 26319 . 2 (𝐺 ∈ UPGraph → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
43ibi 259 1 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  {crab 3093  cdif 3766  c0 4115  𝒫 cpw 4349  {csn 4368   class class class wbr 4843  dom cdm 5312  wf 6097  cfv 6101  cle 10364  2c2 11368  chash 13370  Vtxcvtx 26231  iEdgciedg 26232  UPGraphcupgr 26315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-upgr 26317
This theorem is referenced by:  upgrfn  26322  upgrss  26323  upgrop  26329  upgruhgr  26337  upgrun  26353  umgrislfupgr  26358  upgredgss  26367  edgupgr  26369  upgredg  26373  upgrreslem  26538  upgrres1  26547
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