Theorem nbupgruvtxres 40634

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 4859	. . . . . . 7 ⊢ ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩ ∈ V
3	1, 2	eqeltri 2684	. . . . . 6 ⊢ 𝑆 ∈ V
4		eqid 2610	. . . . . . 7 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
5	4	nbgrssovtx 40586	. . . . . 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 4275	. . . . . 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 40530	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
12	11	eqcomi 2619	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
13	12	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁}) = (Vtx‘𝑆))
14	13	difeq1d 3689	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
15	7, 14	syl5eq 2656	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁, 𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
16	6, 15	sseqtr4d 3605	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
17	16	adantr 480	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
18		simpl 472	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})))
19	18	anim1i 590	. . . . . . 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 1033	. . . . . . 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 223	. . . . . 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 3850	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
23	7, 22	eqtri 2632	. . . . . . . . . . . 12 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
24	23	eleq2i 2680	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}))
25		eldifsn 4260	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛 ≠ 𝑁))
26	24, 25	bitri 263	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛 ≠ 𝑁))
27	26	simplbi 475	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝑉 ∖ {𝐾}))
28		eleq2 2677	. . . . . . . . 9 ⊢ ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑛 ∈ (𝑉 ∖ {𝐾})))
29	27, 28	syl5ibr 235	. . . . . . . 8 ⊢ ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
30	29	adantl 481	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
31	30	imp 444	. . . . . 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 40592	. . . . . 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 63	. . . . 5 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾))
34	33	ralrimiva 2949	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ∀𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})𝑛 ∈ (𝑆 NeighbVtx 𝐾))
35		dfss3 3558	. . . 4 ⊢ ((𝑉 ∖ {𝑁, 𝐾}) ⊆ (𝑆 NeighbVtx 𝐾) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})𝑛 ∈ (𝑆 NeighbVtx 𝐾))
36	34, 35	sylibr 223	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑉 ∖ {𝑁, 𝐾}) ⊆ (𝑆 NeighbVtx 𝐾))
37	17, 36	eqssd 3585	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾}))
38	37	ex 449	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾})))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  ⟨cop 4131   I cid 4948   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   UPGraph cupgr 25747  Edgcedga 25792   NeighbVtx cnbgr 40550

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-vtx 25675  df-iedg 25676  df-upgr 25749  df-edga 25793  df-nbgr 40554

This theorem is referenced by:  uvtxupgrres  40635