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Theorem uhgrspan1 40527
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 3700 . 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 40526 . . 3 (iEdg‘𝑆) = (𝐼𝐹)
7 resresdm 40319 . . 3 ((iEdg‘𝑆) = (𝐼𝐹) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
86, 7mp1i 13 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)))
93uhgrfun 25732 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
10 fvelima 6158 . . . . . . 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 2611 . . . . . . . 8 (𝑖 = 𝑗𝑁 = 𝑁)
15 fveq2 6103 . . . . . . . 8 (𝑖 = 𝑗 → (𝐼𝑖) = (𝐼𝑗))
1614, 15neleq12d 2887 . . . . . . 7 (𝑖 = 𝑗 → (𝑁 ∉ (𝐼𝑖) ↔ 𝑁 ∉ (𝐼𝑗)))
1716, 4elrab2 3333 . . . . . 6 (𝑗𝐹 ↔ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)))
18 fvex 6113 . . . . . . . . . 10 (𝐼𝑗) ∈ V
1918a1i 11 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ V)
202, 3uhgrss 25730 . . . . . . . . . 10 ((𝐺 ∈ UHGraph ∧ 𝑗 ∈ dom 𝐼) → (𝐼𝑗) ⊆ 𝑉)
2120ad2ant2r 779 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ⊆ 𝑉)
22 simprr 792 . . . . . . . . 9 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → 𝑁 ∉ (𝐼𝑗))
23 elpwdifsn 40312 . . . . . . . . 9 (((𝐼𝑗) ∈ V ∧ (𝐼𝑗) ⊆ 𝑉𝑁 ∉ (𝐼𝑗)) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
2419, 21, 22, 23syl3anc 1318 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁}))
25 eleq1 2676 . . . . . . . . 9 (𝑐 = (𝐼𝑗) → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2625eqcoms 2618 . . . . . . . 8 ((𝐼𝑗) = 𝑐 → (𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐼𝑗) ∈ 𝒫 (𝑉 ∖ {𝑁})))
2724, 26syl5ibrcom 236 . . . . . . 7 (((𝐺 ∈ UHGraph ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗))) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
2827ex 449 . . . . . 6 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐼𝑁 ∉ (𝐼𝑗)) → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
2917, 28syl5bi 231 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑗𝐹 → ((𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁}))))
3029rexlimdv 3012 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (∃𝑗𝐹 (𝐼𝑗) = 𝑐𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3113, 30syld 46 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑐 ∈ (𝐼𝐹) → 𝑐 ∈ 𝒫 (𝑉 ∖ {𝑁})))
3231ssrdv 3574 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))
33 opex 4859 . . . . 5 ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩ ∈ V
345, 33eqeltri 2684 . . . 4 𝑆 ∈ V
3534a1i 11 . . 3 (𝑁𝑉𝑆 ∈ V)
362, 3, 4, 5uhgrspan1lem2 40525 . . . . 5 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
3736eqcomi 2619 . . . 4 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
38 eqid 2610 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
396rneqi 5273 . . . . 5 ran (iEdg‘𝑆) = ran (𝐼𝐹)
40 edgaval 25794 . . . . . 6 (𝑆 ∈ V → (Edg‘𝑆) = ran (iEdg‘𝑆))
4134, 40ax-mp 5 . . . . 5 (Edg‘𝑆) = ran (iEdg‘𝑆)
42 df-ima 5051 . . . . 5 (𝐼𝐹) = ran (𝐼𝐹)
4339, 41, 423eqtr4ri 2643 . . . 4 (𝐼𝐹) = (Edg‘𝑆)
4437, 2, 38, 3, 43issubgr 40495 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
4535, 44sylan2 490 . 2 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → (𝑆 SubGraph 𝐺 ↔ ((𝑉 ∖ {𝑁}) ⊆ 𝑉 ∧ (iEdg‘𝑆) = (𝐼 ↾ dom (iEdg‘𝑆)) ∧ (𝐼𝐹) ⊆ 𝒫 (𝑉 ∖ {𝑁}))))
461, 8, 32, 45mpbir3and 1238 1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑉) → 𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wnel 2781  wrex 2897  {crab 2900  Vcvv 3173  cdif 3537  wss 3540  𝒫 cpw 4108  {csn 4125  cop 4131   class class class wbr 4583  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Fun wfun 5798  cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792   SubGraph csubgr 40491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-1st 7059  df-2nd 7060  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-edga 25793  df-subgr 40492
This theorem is referenced by: (None)
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