Theorem lfuhgr 35505

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 29336	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		lfuhgr.2	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	rneqi 5925	. . . . 5 ⊢ ran 𝐼 = ran (iEdg‘𝐺)
4	1, 3	eqtr4i 2795	. . . 4 ⊢ (Edg‘𝐺) = ran 𝐼
5	4	sseq1i 3973	. . 3 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6	2	uhgrfun 29353	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
7		fdmrn 6735	. . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼:dom 𝐼⟶ran 𝐼)
8		fss 6720	. . . . . . 7 ⊢ ((𝐼:dom 𝐼⟶ran 𝐼 ∧ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
9	8	ex 417	. . . . . 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 220	. . . . 5 ⊢ (Fun 𝐼 → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
11	6, 10	syl 18	. . . 4 ⊢ (𝐺 ∈ UHGraph → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
12		frn 6711	. . . 4 ⊢ (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
13	11, 12	impbid1 228	. . 3 ⊢ (𝐺 ∈ UHGraph → (ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
14	5, 13	bitrid 286	. 2 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}))
15		uhgredgss 29418	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
16	15	difss2d 4101	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
17		lfuhgr.1	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
18	17	pweqi 4580	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
19	16, 18	sseqtrrdi 3986	. . 3 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 𝑉)
20		ssrab 4033	. . . 4 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((Edg‘𝐺) ⊆ 𝒫 𝑉 ∧ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
21	20	baib 544	. . 3 ⊢ ((Edg‘𝐺) ⊆ 𝒫 𝑉 → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
22	19, 21	syl 18	. 2 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
23	14, 22	bitr3d 284	1 ⊢ (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4564  {csn 4591   class class class wbr 5110  dom cdm 5659  ran crn 5660  Fun wfun 6528  ⟶wf 6530  ‘cfv 6534   ≤ cle 11240  2c2 12291  ♯chash 14362  Vtxcvtx 29283  iEdgciedg 29284  Edgcedg 29334  UHGraphcuhgr 29343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-edg 29335  df-uhgr 29345

This theorem is referenced by:  lfuhgr2  35506