Theorem lfuhgr 35140

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 29028	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		lfuhgr.2	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	rneqi 5917	. . . . 5 ⊢ ran 𝐼 = ran (iEdg‘𝐺)
4	1, 3	eqtr4i 2761	. . . 4 ⊢ (Edg‘𝐺) = ran 𝐼
5	4	sseq1i 3987	. . 3 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6	2	uhgrfun 29045	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
7		fdmrn 6737	. . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼:dom 𝐼⟶ran 𝐼)
8		fss 6722	. . . . . . 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 6713	. . . 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 29110	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
16	15	difss2d 4114	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
17		lfuhgr.1	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
18	17	pweqi 4591	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
19	16, 18	sseqtrrdi 4000	. . 3 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 𝑉)
20		ssrab 4048	. . . 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 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  {csn 4601   class class class wbr 5119  dom cdm 5654  ran crn 5655  Fun wfun 6525  ⟶wf 6527  ‘cfv 6531   ≤ cle 11270  2c2 12295  ♯chash 14348  Vtxcvtx 28975  iEdgciedg 28976  Edgcedg 29026  UHGraphcuhgr 29035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-edg 29027  df-uhgr 29037

This theorem is referenced by:  lfuhgr2  35141