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Theorem lfuhgr 32792
Description: A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
Hypotheses
Ref Expression
lfuhgr.1 𝑉 = (Vtx‘𝐺)
lfuhgr.2 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
lfuhgr (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉
Allowed substitution hint:   𝐼(𝑥)

Proof of Theorem lfuhgr
StepHypRef Expression
1 edgval 27140 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 lfuhgr.2 . . . . . 6 𝐼 = (iEdg‘𝐺)
32rneqi 5806 . . . . 5 ran 𝐼 = ran (iEdg‘𝐺)
41, 3eqtr4i 2768 . . . 4 (Edg‘𝐺) = ran 𝐼
54sseq1i 3929 . . 3 ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
62uhgrfun 27157 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐼)
7 fdmrn 6577 . . . . . 6 (Fun 𝐼𝐼:dom 𝐼⟶ran 𝐼)
8 fss 6562 . . . . . . 7 ((𝐼:dom 𝐼⟶ran 𝐼 ∧ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
98ex 416 . . . . . 6 (𝐼:dom 𝐼⟶ran 𝐼 → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
107, 9sylbi 220 . . . . 5 (Fun 𝐼 → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
116, 10syl 17 . . . 4 (𝐺 ∈ UHGraph → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
12 frn 6552 . . . 4 (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
1311, 12impbid1 228 . . 3 (𝐺 ∈ UHGraph → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
145, 13syl5bb 286 . 2 (𝐺 ∈ UHGraph → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
15 uhgredgss 27222 . . . . 5 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
1615difss2d 4049 . . . 4 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
17 lfuhgr.1 . . . . 5 𝑉 = (Vtx‘𝐺)
1817pweqi 4531 . . . 4 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
1916, 18sseqtrrdi 3952 . . 3 (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 𝑉)
20 ssrab 3986 . . . 4 ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((Edg‘𝐺) ⊆ 𝒫 𝑉 ∧ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
2120baib 539 . . 3 ((Edg‘𝐺) ⊆ 𝒫 𝑉 → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
2219, 21syl 17 . 2 (𝐺 ∈ UHGraph → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
2314, 22bitr3d 284 1 (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wral 3061  {crab 3065  wss 3866  c0 4237  𝒫 cpw 4513  {csn 4541   class class class wbr 5053  dom cdm 5551  ran crn 5552  Fun wfun 6374  wf 6376  cfv 6380  cle 10868  2c2 11885  chash 13896  Vtxcvtx 27087  iEdgciedg 27088  Edgcedg 27138  UHGraphcuhgr 27147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-edg 27139  df-uhgr 27149
This theorem is referenced by:  lfuhgr2  32793
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