Theorem upgrf 26873

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 26874 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

Step	Hyp	Ref	Expression
1		isupgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		isupgr.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	isupgr 26871	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4	3	ibi 269	1 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  {crab 3144   ∖ cdif 3935  ∅c0 4293  𝒫 cpw 4541  {csn 4569   class class class wbr 5068  dom cdm 5557  ⟶wf 6353  ‘cfv 6357   ≤ cle 10678  2c2 11695  ♯chash 13693  Vtxcvtx 26783  iEdgciedg 26784  UPGraphcupgr 26867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-upgr 26869

This theorem is referenced by:  upgrfn  26874  upgrss  26875  upgrop  26881  upgruhgr  26889  upgrun  26905  umgrislfupgr  26910  upgredgss  26919  edgupgr  26921  upgredg  26924  upgrreslem  27088  upgrres1  27097