Theorem lfuhgr 29783

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 29684	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		lfuhgr.2	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	rneqi 5931	. . . . 5 ⊢ ran 𝐼 = ran (iEdg‘𝐺)
4	1, 3	eqtr4i 2789	. . . 4 ⊢ (Edg‘𝐺) = ran 𝐼
5	4	sseq1i 3941	. . 3 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6	2	uhgrfun 29701	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
7		fdmrn 6771	. . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼:dom 𝐼⟶ran 𝐼)
8		fss 6756	. . . . . . 7 ⊢ ((𝐼:dom 𝐼⟶ran 𝐼 ∧ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
9	8	ex 418	. . . . . 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 6747	. . . 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 29766	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
16	15	difss2d 4078	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
17		lfuhgr.1	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
18	17	pweqi 4576	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
19	16, 18	sseqtrrdi 3954	. . 3 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 𝑉)
20		ssrab 3999	. . . 4 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((Edg‘𝐺) ⊆ 𝒫 𝑉 ∧ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
21	20	baib 545	. . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5108  dom cdm 5658  ran crn 5659  Fun wfun 6560  ⟶wf 6562  ‘cfv 6566   ≤ cle 11383  2c2 12437  ♯chash 14515  Vtxcvtx 29631  iEdgciedg 29632  Edgcedg 29682  UHGraphcuhgr 29691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-edg 29683  df-uhgr 29693

This theorem is used by:  lfuhgr2  29784