Theorem uhgrspan1 29230

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 4100	. 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 29229	. . 3 ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹)
7		resresdm 6206	. . 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 28993	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
10		fvelima 6926	. . . . . . 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 2730	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → 𝑁 = 𝑁)
15		fveq2 6858	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝐼‘𝑖) = (𝐼‘𝑗))
16	14, 15	neleq12d 3034	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (𝑁 ∉ (𝐼‘𝑖) ↔ 𝑁 ∉ (𝐼‘𝑗)))
17	16, 4	elrab2 3662	. . . . . 6 ⊢ (𝑗 ∈ 𝐹 ↔ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗)))
18		fvexd 6873	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ V)
19	2, 3	uhgrss 28991	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼‘𝑗) ⊆ 𝑉)
20	19	ad2ant2r 747	. . . . . . . . 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 4753	. . . . . . . . 9 ⊢ (((𝐼‘𝑗) ∈ V ∧ (𝐼‘𝑗) ⊆ 𝑉 ∧ 𝑁 ∉ (𝐼‘𝑗)) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
23	18, 20, 21, 22	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐼 ∧ 𝑁 ∉ (𝐼‘𝑗))) → (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
24		eleq1 2816	. . . . . . . . 9 ⊢ (𝑐 = (𝐼‘𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼‘𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
25	24	eqcoms 2737	. . . . . . . 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 3132	. . . 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 3952	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → (𝐼 “ 𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
32		opex 5424	. . . . 5 ⊢ ⟨(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)⟩ ∈ V
33	5, 32	eqeltri 2824	. . . 4 ⊢ 𝑆 ∈ V
34	33	a1i 11	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ∈ V)
35	2, 3, 4, 5	uhgrspan1lem2 29228	. . . . 5 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
36	35	eqcomi 2738	. . . 4 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
37		eqid 2729	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
38	6	rneqi 5901	. . . . 5 ⊢ ran (iEdg‘𝑆) = ran (𝐼 ↾ 𝐹)
39		edgval 28976	. . . . 5 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
40		df-ima 5651	. . . . 5 ⊢ (𝐼 “ 𝐹) = ran (𝐼 ↾ 𝐹)
41	38, 39, 40	3eqtr4ri 2763	. . . 4 ⊢ (𝐼 “ 𝐹) = (Edg‘𝑆)
42	36, 2, 37, 3, 41	issubgr 29198	. . 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 593	. 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 1343	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  𝒫 cpw 4563  {csn 4589  ⟨cop 4595   class class class wbr 5107  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Fun wfun 6505  ‘cfv 6511  Vtxcvtx 28923  iEdgciedg 28924  Edgcedg 28974  UHGraphcuhgr 28983   SubGraph csubgr 29194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-1st 7968  df-2nd 7969  df-vtx 28925  df-iedg 28926  df-edg 28975  df-uhgr 28985  df-subgr 29195

This theorem is referenced by: (None)