Theorem upgredgss 26247

Description: The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020.)

Assertion

Ref	Expression
upgredgss	⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Distinct variable group:   𝑥,𝐺

Proof of Theorem upgredgss

Step	Hyp	Ref	Expression
1		edgval 26161	. 2 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		eqid 2760	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2760	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgrf 26201	. . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
5		frn 6214	. . 3 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6	4, 5	syl 17	. 2 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
7	1, 6	syl5eqss 3790	1 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Syntax hints:   → wi 4   ∈ wcel 2139  {crab 3054   ∖ cdif 3712   ⊆ wss 3715  ∅c0 4058  𝒫 cpw 4302  {csn 4321   class class class wbr 4804  dom cdm 5266  ran crn 5267  ⟶wf 6045  ‘cfv 6049   ≤ cle 10287  2c2 11282  ♯chash 13331  Vtxcvtx 26094  iEdgciedg 26095  Edgcedg 26159  UPGraphcupgr 26195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-edg 26160  df-upgr 26197

This theorem is referenced by:  uspgrupgrushgr  26292  upgredgssspr  42279