Theorem lfuhgr 35154

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

Step	Hyp	Ref	Expression
1		edgval 29022	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		lfuhgr.2	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	rneqi 5872	. . . . 5 ⊢ ran 𝐼 = ran (iEdg‘𝐺)
4	1, 3	eqtr4i 2757	. . . 4 ⊢ (Edg‘𝐺) = ran 𝐼
5	4	sseq1i 3958	. . 3 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6	2	uhgrfun 29039	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
7		fdmrn 6677	. . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼:dom 𝐼⟶ran 𝐼)
8		fss 6662	. . . . . . 7 ⊢ ((𝐼:dom 𝐼⟶ran 𝐼 ∧ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
9	8	ex 412	. . . . . 6 ⊢ (𝐼:dom 𝐼⟶ran 𝐼 → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
10	7, 9	sylbi 217	. . . . 5 ⊢ (Fun 𝐼 → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
11	6, 10	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
12		frn 6653	. . . 4 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
13	11, 12	impbid1 225	. . 3 ⊢ (𝐺 ∈ UHGraph → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
14	5, 13	bitrid 283	. 2 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
15		uhgredgss 29104	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
16	15	difss2d 4084	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
17		lfuhgr.1	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
18	17	pweqi 4561	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
19	16, 18	sseqtrrdi 3971	. . 3 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 𝑉)
20		ssrab 4018	. . . 4 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((Edg‘𝐺) ⊆ 𝒫 𝑉 ∧ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
21	20	baib 535	. . 3 ⊢ ((Edg‘𝐺) ⊆ 𝒫 𝑉 → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
22	19, 21	syl 17	. 2 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
23	14, 22	bitr3d 281	1 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3897  ∅c0 4278  𝒫 cpw 4545  {csn 4571   class class class wbr 5086  dom cdm 5611  ran crn 5612  Fun wfun 6470  ⟶wf 6472  ‘cfv 6476   ≤ cle 11142  2c2 12175  ♯chash 14232  Vtxcvtx 28969  iEdgciedg 28970  Edgcedg 29020  UHGraphcuhgr 29029

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-edg 29021  df-uhgr 29031

This theorem is referenced by:  lfuhgr2  35155