Theorem upgrf 29646

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 29647 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 29644	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
4	3	ibi 270	1 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3413   ∖ cdif 3896  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103  dom cdm 5651  ⟶wf 6527  ‘cfv 6531   ≤ cle 11325  2c2 12378  ♯chash 14454  Vtxcvtx 29556  iEdgciedg 29557  UPGraphcupgr 29640

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-upgr 29642

This theorem is used by:  upgrfn  29647  upgrss  29648  upgrop  29654  upgruhgr  29662  upgrun  29678  umgrislfupgr  29683  upgredgss  29692  edgupgr  29694  upgredg  29697  upgrreslem  29867  upgrres1  29876