Theorem lfuhgr 29726

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 29627	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		lfuhgr.2	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	rneqi 5918	. . . . 5 ⊢ ran 𝐼 = ran (iEdg‘𝐺)
4	1, 3	eqtr4i 2786	. . . 4 ⊢ (Edg‘𝐺) = ran 𝐼
5	4	sseq1i 3936	. . 3 ⊢ ((Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ran 𝐼 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6	2	uhgrfun 29644	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
7		fdmrn 6741	. . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼:dom 𝐼⟶ran 𝐼)
8		fss 6726	. . . . . . 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 6717	. . . 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 29709	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
16	15	difss2d 4075	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
17		lfuhgr.1	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
18	17	pweqi 4572	. . . 4 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
19	16, 18	sseqtrrdi 3949	. . 3 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ 𝒫 𝑉)
20		ssrab 3996	. . . 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 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  {csn 4583   class class class wbr 5102  dom cdm 5650  ran crn 5651  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538   ≤ cle 11344  2c2 12397  ♯chash 14474  Vtxcvtx 29574  iEdgciedg 29575  Edgcedg 29625  UHGraphcuhgr 29634

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-edg 29626  df-uhgr 29636

This theorem is used by:  lfuhgr2  29727