Theorem upgrf 27359

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 27360 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 27357	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4	3	ibi 266	1 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3067   ∖ cdif 3880  ∅c0 4253  𝒫 cpw 4530  {csn 4558   class class class wbr 5070  dom cdm 5580  ⟶wf 6414  ‘cfv 6418   ≤ cle 10941  2c2 11958  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  UPGraphcupgr 27353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-upgr 27355

This theorem is referenced by:  upgrfn  27360  upgrss  27361  upgrop  27367  upgruhgr  27375  upgrun  27391  umgrislfupgr  27396  upgredgss  27405  edgupgr  27407  upgredg  27410  upgrreslem  27574  upgrres1  27583