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Theorem uhgrspan1 29160
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 4125 . 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 29159 . . 3 (iEdg‘𝑆) = (𝐼𝐹)
7 resresdm 6232 . . 3 ((iEdg‘𝑆) = (𝐼𝐹) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
86, 7mp1i 13 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
93uhgrfun 28923 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
10 fvelima 6959 . . . . . . 7 ((Fun 𝐼𝑐 ∈ (𝐼𝐹)) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐)
1110ex 411 . . . . . 6 (Fun 𝐼 → (𝑐 ∈ (𝐼𝐹) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐))
129, 11syl 17 . . . . 5 (𝐺 ∈ UHGraph → (𝑐 ∈ (𝐼𝐹) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐))
1312adantr 479 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑐 ∈ (𝐼𝐹) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐))
14 eqidd 2726 . . . . . . . 8 (𝑖 = 𝑗𝑁 = 𝑁)
15 fveq2 6892 . . . . . . . 8 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
1614, 15neleq12d 3041 . . . . . . 7 (𝑖 = 𝑗 → (𝑁 ∉ (𝐼𝑖) ↔ 𝑁 ∉ (𝐼𝑗)))
1716, 4elrab2 3677 . . . . . 6 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)))
18 fvexd 6907 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ V)
192, 3uhgrss 28921 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼𝑗) ⊆ 𝑉)
2019ad2ant2r 745 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ⊆ 𝑉)
21 simprr 771 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → 𝑁 ∉ (𝐼𝑗))
22 elpwdifsn 4788 . . . . . . . . 9 (((𝐼𝑗) ∈ V ∧ (𝐼𝑗) ⊆ 𝑉𝑁 ∉ (𝐼𝑗)) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2318, 20, 21, 22syl3anc 1368 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
24 eleq1 2813 . . . . . . . . 9 (𝑐 = (𝐼𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2524eqcoms 2733 . . . . . . . 8 ((𝐼𝑗) = 𝑐 → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2623, 25syl5ibrcom 246 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
2726ex 411 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
2817, 27biimtrid 241 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑗𝐹 → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
2928rexlimdv 3143 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (∃𝑗𝐹 (𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3013, 29syld 47 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑐 ∈ (𝐼𝐹) → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3130ssrdv 3978 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
32 opex 5460 . . . . 5 ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩ ∈ V
335, 32eqeltri 2821 . . . 4 𝑆 ∈ V
3433a1i 11 . . 3 (𝑁𝑉𝑆 ∈ V)
352, 3, 4, 5uhgrspan1lem2 29158 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
3635eqcomi 2734 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
37 eqid 2725 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
386rneqi 5933 . . . . 5 ran (iEdg‘𝑆) = ran (𝐼𝐹)
39 edgval 28906 . . . . 5 (Edg‘𝑆) = ran (iEdg‘𝑆)
40 df-ima 5685 . . . . 5 (𝐼𝐹) = ran (𝐼𝐹)
4138, 39, 403eqtr4ri 2764 . . . 4 (𝐼𝐹) = (Edg‘𝑆)
4236, 2, 37, 3, 41issubgr 29128 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
4334, 42sylan2 591 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
441, 8, 31, 43mpbir3and 1339 1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wnel 3036  wrex 3060  {crab 3419  Vcvv 3463  cdif 3936  wss 3939  𝒫 cpw 4598  {csn 4624  cop 4630   class class class wbr 5143  dom cdm 5672  ran crn 5673  cres 5674  cima 5675  Fun wfun 6537  cfv 6543  Vtxcvtx 28853  iEdgciedg 28854  Edgcedg 28904  UHGraphcuhgr 28913   SubGraph csubgr 29124
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-1st 7991  df-2nd 7992  df-vtx 28855  df-iedg 28856  df-edg 28905  df-uhgr 28915  df-subgr 29125
This theorem is referenced by: (None)
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