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Theorem uhgrspan1 26240
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 3771 . 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 26239 . . 3 (iEdg‘𝑆) = (𝐼𝐹)
7 resresdm 5664 . . 3 ((iEdg‘𝑆) = (𝐼𝐹) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
86, 7mp1i 13 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
93uhgrfun 26006 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
10 fvelima 6287 . . . . . . 7 ((Fun 𝐼𝑐 ∈ (𝐼𝐹)) → ∃𝑗𝐹 (𝐼𝑗) = 𝑐)
1110ex 449 . . . . . 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 2652 . . . . . . . 8 (𝑖 = 𝑗𝑁 = 𝑁)
15 fveq2 6229 . . . . . . . 8 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
1614, 15neleq12d 2930 . . . . . . 7 (𝑖 = 𝑗 → (𝑁 ∉ (𝐼𝑖) ↔ 𝑁 ∉ (𝐼𝑗)))
1716, 4elrab2 3399 . . . . . 6 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)))
18 fvexd 6241 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ V)
192, 3uhgrss 26004 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼𝑗) ⊆ 𝑉)
2019ad2ant2r 798 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ⊆ 𝑉)
21 simprr 811 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → 𝑁 ∉ (𝐼𝑗))
22 elpwdifsn 4352 . . . . . . . . 9 (((𝐼𝑗) ∈ V ∧ (𝐼𝑗) ⊆ 𝑉𝑁 ∉ (𝐼𝑗)) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2318, 20, 21, 22syl3anc 1366 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
24 eleq1 2718 . . . . . . . . 9 (𝑐 = (𝐼𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2524eqcoms 2659 . . . . . . . 8 ((𝐼𝑗) = 𝑐 → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2623, 25syl5ibrcom 237 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
2726ex 449 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
2817, 27syl5bi 232 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑗𝐹 → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
2928rexlimdv 3059 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (∃𝑗𝐹 (𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3013, 29syld 47 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑐 ∈ (𝐼𝐹) → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3130ssrdv 3642 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
32 opex 4962 . . . . 5 ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩ ∈ V
335, 32eqeltri 2726 . . . 4 𝑆 ∈ V
3433a1i 11 . . 3 (𝑁𝑉𝑆 ∈ V)
352, 3, 4, 5uhgrspan1lem2 26238 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
3635eqcomi 2660 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
37 eqid 2651 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
386rneqi 5384 . . . . 5 ran (iEdg‘𝑆) = ran (𝐼𝐹)
39 edgval 25986 . . . . 5 (Edg‘𝑆) = ran (iEdg‘𝑆)
40 df-ima 5156 . . . . 5 (𝐼𝐹) = ran (𝐼𝐹)
4138, 39, 403eqtr4ri 2684 . . . 4 (𝐼𝐹) = (Edg‘𝑆)
4236, 2, 37, 3, 41issubgr 26208 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
4334, 42sylan2 490 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
441, 8, 31, 43mpbir3and 1264 1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wnel 2926  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  𝒫 cpw 4191  {csn 4210  cop 4216   class class class wbr 4685  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UHGraphcuhgr 25996   SubGraph csubgr 26204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-1st 7210  df-2nd 7211  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-subgr 26205
This theorem is referenced by: (None)
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