Theorem uhgrspan1 29068

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 4127	. 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 29067	. . 3 ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹)
7		resresdm 6226	. . 3 ⊢ ((iEdg‘𝑆) = (𝐼 ↾ 𝐹) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
8	6, 7	mp1i 13	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
9	3	uhgrfun 28834	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
10		fvelima 6951	. . . . . . 7 ⊢ ((Fun 𝐼 ∧ 𝑐 ∈ (𝐼 “ 𝐹)) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐)
11	10	ex 412	. . . . . 6 ⊢ (Fun 𝐼 → (𝑐 ∈ (𝐼 “ 𝐹) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐))
12	9, 11	syl 17	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (𝑐 ∈ (𝐼 “ 𝐹) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐))
13	12	adantr 480	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑐 ∈ (𝐼 “ 𝐹) → ∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐))
14		eqidd 2727	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → 𝑁 = 𝑁)
15		fveq2 6885	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
16	14, 15	neleq12d 3045	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑁 ∉ (𝐼‘𝑖) ↔ 𝑁 ∉ (𝐼‘𝑗)))
17	16, 4	elrab2 3681	. . . . . 6 ⊢ (𝑗 ∈ 𝐹 ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗)))
18		fvexd 6900	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ V)
19	2, 3	uhgrss 28832	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ⊆ 𝑉)
20	19	ad2ant2r 744	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ⊆ 𝑉)
21		simprr 770	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → 𝑁 ∉ (𝐼‘𝑗))
22		elpwdifsn 4787	. . . . . . . . 9 ⊢ (((𝐼‘𝑗) ∈ V ∧ (𝐼‘𝑗) ⊆ 𝑉 ∧ 𝑁 ∉ (𝐼‘𝑗)) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
23	18, 20, 21, 22	syl3anc 1368	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
24		eleq1 2815	. . . . . . . . 9 ⊢ (𝑐 = (𝐼‘𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
25	24	eqcoms 2734	. . . . . . . 8 ⊢ ((𝐼‘𝑗) = 𝑐 → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
26	23, 25	syl5ibrcom 246	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → ((𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
27	26	ex 412	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗)) → ((𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
28	17, 27	biimtrid 241	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ 𝐹 → ((𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
29	28	rexlimdv 3147	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑗 ∈ 𝐹 (𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
30	13, 29	syld 47	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑐 ∈ (𝐼 “ 𝐹) → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
31	30	ssrdv 3983	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ 𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
32		opex 5457	. . . . 5 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)⟩ ∈ V
33	5, 32	eqeltri 2823	. . . 4 ⊢ 𝑆 ∈ V
34	33	a1i 11	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ∈ V)
35	2, 3, 4, 5	uhgrspan1lem2 29066	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
36	35	eqcomi 2735	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
37		eqid 2726	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
38	6	rneqi 5930	. . . . 5 ⊢ ran (iEdg‘𝑆) = ran (𝐼 ↾ 𝐹)
39		edgval 28817	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
40		df-ima 5682	. . . . 5 ⊢ (𝐼 “ 𝐹) = ran (𝐼 ↾ 𝐹)
41	38, 39, 40	3eqtr4ri 2765	. . . 4 ⊢ (𝐼 “ 𝐹) = (Edg‘𝑆)
42	36, 2, 37, 3, 41	issubgr 29036	. . 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 592	. 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 1339	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∉ wnel 3040  ∃wrex 3064  {crab 3426  Vcvv 3468   ∖ cdif 3940   ⊆ wss 3943  𝒫 cpw 4597  {csn 4623  ⟨cop 4629   class class class wbr 5141  dom cdm 5669  ran crn 5670   ↾ cres 5671   “ cima 5672  Fun wfun 6531  ‘cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  UHGraphcuhgr 28824   SubGraph csubgr 29032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-1st 7974  df-2nd 7975  df-vtx 28766  df-iedg 28767  df-edg 28816  df-uhgr 28826  df-subgr 29033

This theorem is referenced by: (None)