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Theorem uhgrspan1 29320
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 𝐼𝑁 ∉ (𝐼𝑖)}
uhgrspan1.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩
Assertion
Ref Expression
uhgrspan1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 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.
StepHypRef Expression
1 difssd 4137 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑉 ∖ {𝑁}) ⊆ 𝑉)
2 uhgrspan1.v . . . 4 𝑉 = (Vtx‘𝐺)
3 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
4 uhgrspan1.f . . . 4 𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}
5 uhgrspan1.s . . . 4 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩
62, 3, 4, 5uhgrspan1lem3 29319 . . 3 (iEdg‘𝑆) = (𝐼𝐹)
7 resresdm 6253 . . 3 ((iEdg‘𝑆) = (𝐼𝐹) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
86, 7mp1i 13 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
93uhgrfun 29083 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
10 fvelima 6974 . . . . . . 7 ((Fun 𝐼𝑐 ∈ (𝐼𝐹)) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐)
1110ex 412 . . . . . 6 (Fun 𝐼 → (𝑐 ∈ (𝐼𝐹) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐))
129, 11syl 17 . . . . 5 (𝐺 ∈ UHGraph → (𝑐 ∈ (𝐼𝐹) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐))
1312adantr 480 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑐 ∈ (𝐼𝐹) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐))
14 eqidd 2738 . . . . . . . 8 (𝑖 = 𝑗𝑁 = 𝑁)
15 fveq2 6906 . . . . . . . 8 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
1614, 15neleq12d 3051 . . . . . . 7 (𝑖 = 𝑗 → (𝑁 ∉ (𝐼𝑖) ↔ 𝑁 ∉ (𝐼𝑗)))
1716, 4elrab2 3695 . . . . . 6 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)))
18 fvexd 6921 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ V)
192, 3uhgrss 29081 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼𝑗) ⊆ 𝑉)
2019ad2ant2r 747 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ⊆ 𝑉)
21 simprr 773 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → 𝑁 ∉ (𝐼𝑗))
22 elpwdifsn 4789 . . . . . . . . 9 (((𝐼𝑗) ∈ V ∧ (𝐼𝑗) ⊆ 𝑉𝑁 ∉ (𝐼𝑗)) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2318, 20, 21, 22syl3anc 1373 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
24 eleq1 2829 . . . . . . . . 9 (𝑐 = (𝐼𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2524eqcoms 2745 . . . . . . . 8 ((𝐼𝑗) = 𝑐 → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2623, 25syl5ibrcom 247 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
2726ex 412 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
2817, 27biimtrid 242 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑗𝐹 → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
2928rexlimdv 3153 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (∃𝑗𝐹 (𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3013, 29syld 47 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑐 ∈ (𝐼𝐹) → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3130ssrdv 3989 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
32 opex 5469 . . . . 5 ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩ ∈ V
335, 32eqeltri 2837 . . . 4 𝑆 ∈ V
3433a1i 11 . . 3 (𝑁𝑉𝑆 ∈ V)
352, 3, 4, 5uhgrspan1lem2 29318 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
3635eqcomi 2746 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
37 eqid 2737 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
386rneqi 5948 . . . . 5 ran (iEdg‘𝑆) = ran (𝐼𝐹)
39 edgval 29066 . . . . 5 (Edg‘𝑆) = ran (iEdg‘𝑆)
40 df-ima 5698 . . . . 5 (𝐼𝐹) = ran (𝐼𝐹)
4138, 39, 403eqtr4ri 2776 . . . 4 (𝐼𝐹) = (Edg‘𝑆)
4236, 2, 37, 3, 41issubgr 29288 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
4334, 42sylan2 593 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
441, 8, 31, 43mpbir3and 1343 1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wnel 3046  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  wss 3951  𝒫 cpw 4600  {csn 4626  cop 4632   class class class wbr 5143  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073   SubGraph csubgr 29284
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1st 8014  df-2nd 8015  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-subgr 29285
This theorem is referenced by: (None)
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