Theorem nbupgruvtxres 26189

Description: The neighborhood of a universal vertex in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)

Hypotheses

Ref	Expression
nbupgruvtxres.v	⊢ 𝑉 = (Vtx‘𝐺)
nbupgruvtxres.e	⊢ 𝐸 = (Edg‘𝐺)
nbupgruvtxres.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
nbupgruvtxres.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩

Assertion

Ref	Expression
nbupgruvtxres	⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾})))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝐾   𝑒,𝑁   𝑒,𝑉

Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem nbupgruvtxres

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbupgruvtxres.s	. . . . . . 7 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
2		opex 4898	. . . . . . 7 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
3	1, 2	eqeltri 2700	. . . . . 6 ⊢ 𝑆 ∈ V
4		eqid 2626	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
5	4	nbgrssovtx 26141	. . . . . 6 ⊢ (𝑆 ∈ V → (𝑆 NeighbVtx 𝐾) ⊆ ((Vtx‘𝑆) ∖ {𝐾}))
6	3, 5	mp1i 13	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑆 NeighbVtx 𝐾) ⊆ ((Vtx‘𝑆) ∖ {𝐾}))
7		difpr 4308	. . . . . 6 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
8		nbupgruvtxres.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
9		nbupgruvtxres.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
10		nbupgruvtxres.f	. . . . . . . . . 10 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
11	8, 9, 10, 1	upgrres1lem2 26085	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
12	11	eqcomi 2635	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
13	12	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁}) = (Vtx‘𝑆))
14	13	difeq1d 3710	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
15	7, 14	syl5eq 2672	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁, 𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
16	6, 15	sseqtr4d 3626	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
17	16	adantr 481	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
18		simpl 473	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})))
19	18	anim1i 591	. . . . . . 7 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
20		df-3an 1038	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) ↔ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
21	19, 20	sylibr 224	. . . . . 6 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
22		dif32 3872	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
23	7, 22	eqtri 2648	. . . . . . . . . . . 12 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
24	23	eleq2i 2696	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}))
25		eldifsn 4292	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛 ≠ 𝑁))
26	24, 25	bitri 264	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛 ≠ 𝑁))
27	26	simplbi 476	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝑉 ∖ {𝐾}))
28		eleq2 2693	. . . . . . . . 9 ⊢ ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑛 ∈ (𝑉 ∖ {𝐾})))
29	27, 28	syl5ibr 236	. . . . . . . 8 ⊢ ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
30	29	adantl 482	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
31	30	imp 445	. . . . . 6 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾))
32	8, 9, 10, 1	nbupgrres 26147	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾)))
33	21, 31, 32	sylc 65	. . . . 5 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾))
34	33	ralrimiva 2965	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ∀𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})𝑛 ∈ (𝑆 NeighbVtx 𝐾))
35		dfss3 3578	. . . 4 ⊢ ((𝑉 ∖ {𝑁, 𝐾}) ⊆ (𝑆 NeighbVtx 𝐾) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})𝑛 ∈ (𝑆 NeighbVtx 𝐾))
36	34, 35	sylibr 224	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑉 ∖ {𝑁, 𝐾}) ⊆ (𝑆 NeighbVtx 𝐾))
37	17, 36	eqssd 3605	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾}))
38	37	ex 450	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾})))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992   ≠ wne 2796   ∉ wnel 2899  ∀wral 2912  {crab 2916  Vcvv 3191   ∖ cdif 3557   ⊆ wss 3560  {csn 4153  {cpr 4155  ⟨cop 4159   I cid 4989   ↾ cres 5081  ‘cfv 5850  (class class class)co 6605  Vtxcvtx 25769  Edgcedg 25834   UPGraph cupgr 25866   NeighbVtx cnbgr 26105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055  df-vtx 25771  df-iedg 25772  df-edg 25835  df-upgr 25868  df-nbgr 26109

This theorem is referenced by:  uvtxupgrres  26190