Theorem uhgrspan1 29338

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 4160	. 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 29337	. . 3 ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹)
7		resresdm 6264	. . 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 29101	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
10		fvelima 6987	. . . . . . 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 2741	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → 𝑁 = 𝑁)
15		fveq2 6920	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
16	14, 15	neleq12d 3057	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑁 ∉ (𝐼‘𝑖) ↔ 𝑁 ∉ (𝐼‘𝑗)))
17	16, 4	elrab2 3711	. . . . . 6 ⊢ (𝑗 ∈ 𝐹 ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗)))
18		fvexd 6935	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ V)
19	2, 3	uhgrss 29099	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ⊆ 𝑉)
20	19	ad2ant2r 746	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ⊆ 𝑉)
21		simprr 772	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → 𝑁 ∉ (𝐼‘𝑗))
22		elpwdifsn 4814	. . . . . . . . 9 ⊢ (((𝐼‘𝑗) ∈ V ∧ (𝐼‘𝑗) ⊆ 𝑉 ∧ 𝑁 ∉ (𝐼‘𝑗)) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
23	18, 20, 21, 22	syl3anc 1371	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
24		eleq1 2832	. . . . . . . . 9 ⊢ (𝑐 = (𝐼‘𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
25	24	eqcoms 2748	. . . . . . . 8 ⊢ ((𝐼‘𝑗) = 𝑐 → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
26	23, 25	syl5ibrcom 247	. . . . . . 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 242	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ 𝐹 → ((𝐼‘𝑗) = 𝑐 → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
29	28	rexlimdv 3159	. . . 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 4014	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ 𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
32		opex 5484	. . . . 5 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)⟩ ∈ V
33	5, 32	eqeltri 2840	. . . 4 ⊢ 𝑆 ∈ V
34	33	a1i 11	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ∈ V)
35	2, 3, 4, 5	uhgrspan1lem2 29336	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
36	35	eqcomi 2749	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
37		eqid 2740	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
38	6	rneqi 5962	. . . . 5 ⊢ ran (iEdg‘𝑆) = ran (𝐼 ↾ 𝐹)
39		edgval 29084	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
40		df-ima 5713	. . . . 5 ⊢ (𝐼 “ 𝐹) = ran (𝐼 ↾ 𝐹)
41	38, 39, 40	3eqtr4ri 2779	. . . 4 ⊢ (𝐼 “ 𝐹) = (Edg‘𝑆)
42	36, 2, 37, 3, 41	issubgr 29306	. . 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 1342	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∉ wnel 3052  ∃wrex 3076  {crab 3443  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648  ⟨cop 4654   class class class wbr 5166  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6567  ‘cfv 6573  Vtxcvtx 29031  iEdgciedg 29032  Edgcedg 29082  UHGraphcuhgr 29091   SubGraph csubgr 29302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-1st 8030  df-2nd 8031  df-vtx 29033  df-iedg 29034  df-edg 29083  df-uhgr 29093  df-subgr 29303

This theorem is referenced by: (None)