Theorem uhgrspan1 29690

Description: The induced subgraph 𝑆 of a hypergraph 𝐺 obtained by removing one vertex is actually a subgraph of 𝐺. A subgraph is called induced or spanned by a subset of vertices of a graph if it contains all edges of the original graph that join two vertices of the subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
uhgrspan1.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrspan1.i	⊢ 𝐼 = (iEdg‘𝐺)
uhgrspan1.f	⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)}
uhgrspan1.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)⟩

Assertion

Ref	Expression
uhgrspan1	⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 SubGraph 𝐺)

Distinct variable groups:   𝑖,𝐼   𝑖,𝑁

Allowed substitution hints:   𝑆(𝑖)   𝐹(𝑖)   𝐺(𝑖)   𝑉(𝑖)

Proof of Theorem uhgrspan1

Dummy variables 𝑐 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difssd 4094	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) ⊆ 𝑉)
2		uhgrspan1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3		uhgrspan1.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
4		uhgrspan1.f	. . . 4 ⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)}
5		uhgrspan1.s	. . . 4 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)⟩
6	2, 3, 4, 5	uhgrspan1lem3 29689	. . 3 ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹)
7		resresdm 6239	. . 3 ⊢ ((iEdg‘𝑆) = (𝐼 ↾ 𝐹) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
8	6, 7	mp1i 14	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
9	3	uhgrfun 29453	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
10		fvelima 6953	. . . . . . 7 ⊢ ((Fun 𝐼 ∧ 𝑐 ∈ (𝐼 “ 𝐹)) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐)
11	10	ex 418	. . . . . 6 ⊢ (Fun 𝐼 → (𝑐 ∈ (𝐼 “ 𝐹) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐))
12	9, 11	syl 18	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝑐 ∈ (𝐼 “ 𝐹) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐))
13	12	adantr 486	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑐 ∈ (𝐼 “ 𝐹) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐))
14		eqidd 2767	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → 𝑁 = 𝑁)
15		fveq2 6888	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
16	14, 15	neleq12d 3072	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑁 ∉ (𝐼‘𝑖) ↔ 𝑁 ∉ (𝐼‘𝑗)))
17	16, 4	elrab2 3657	. . . . . 6 ⊢ (𝑗 ∈ 𝐹 ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗)))
18		fvexd 6903	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ V)
19	2, 3	uhgrss 29451	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ⊆ 𝑉)
20	19	ad2ant2r 760	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ⊆ 𝑉)
21		simprr 785	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → 𝑁 ∉ (𝐼‘𝑗))
22		elpwdifsn 4762	. . . . . . . . 9 ⊢ (((𝐼‘𝑗) ∈ V ∧ (𝐼‘𝑗) ⊆ 𝑉 ∧ 𝑁 ∉ (𝐼‘𝑗)) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
23	18, 20, 21, 22	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
24		eleq1 2854	. . . . . . . . 9 ⊢ (𝑐 = (𝐼‘𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
25	24	eqcoms 2774	. . . . . . . 8 ⊢ ((𝐼‘𝑗) = 𝑐 → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
26	23, 25	syl5ibrcom 250	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → ((𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
27	26	ex 418	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
28	17, 27	biimtrid 245	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ 𝐹 → ((𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
29	28	rexlimdv 3167	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
30	13, 29	syld 48	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑐 ∈ (𝐼 “ 𝐹) → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
31	30	ssrdv 3946	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ 𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
32		opex 5450	. . . . 5 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)⟩ ∈ V
33	5, 32	eqeltri 2862	. . . 4 ⊢ 𝑆 ∈ V
34	33	a1i 11	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ∈ V)
35	2, 3, 4, 5	uhgrspan1lem2 29688	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
36	35	eqcomi 2775	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
37		eqid 2766	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
38	6	rneqi 5932	. . . . 5 ⊢ ran (iEdg‘𝑆) = ran (𝐼 ↾ 𝐹)
39		edgval 29436	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
40		df-ima 5679	. . . . 5 ⊢ (𝐼 “ 𝐹) = ran (𝐼 ↾ 𝐹)
41	38, 39, 40	3eqtr4ri 2800	. . . 4 ⊢ (𝐼 “ 𝐹) = (Edg‘𝑆)
42	36, 2, 37, 3, 41	issubgr 29658	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼 “ 𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
43	34, 42	sylan2 605	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼 “ 𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
44	1, 8, 31, 43	mpbir3and 1361	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 SubGraph 𝐺)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∉ wnel 3067  ∃wrex 3092  {crab 3419  Vcvv 3458   ∖ cdif 3905   ⊆ wss 3908  𝒫 cpw 4567  {csn 4594  ⟨cop 4600   class class class wbr 5114  dom cdm 5666  ran crn 5667   ↾ cres 5668   “ cima 5669  Fun wfun 6537  ‘cfv 6543  Vtxcvtx 29383  iEdgciedg 29384  Edgcedg 29434  UHGraphcuhgr 29443   SubGraph csubgr 29654

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-1st 7995  df-2nd 7996  df-vtx 29385  df-iedg 29386  df-edg 29435  df-uhgr 29445  df-subgr 29655

This theorem is used by: (None)