Theorem nbupgruvtxres 29853

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.) (Proof shortened by AV, 13-Feb-2022.)

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		eqid 2762	. . . . . 6 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2	1	nbgrssovtx 29807	. . . . 5 ⊢ (𝑆 NeighbVtx 𝐾) ⊆ ((Vtx‘𝑆) ∖ {𝐾})
3		difpr 4769	. . . . . 6 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
4		nbupgruvtxres.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
5		nbupgruvtxres.e	. . . . . . . . . 10 ⊢ 𝐸 = (Edg‘𝐺)
6		nbupgruvtxres.f	. . . . . . . . . 10 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
7		nbupgruvtxres.s	. . . . . . . . . 10 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
8	4, 5, 6, 7	upgrres1lem2 29757	. . . . . . . . 9 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
9	8	eqcomi 2771	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
10	9	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁}) = (Vtx‘𝑆))
11	10	difeq1d 4076	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
12	3, 11	eqtrid 2809	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁, 𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
13	2, 12	sseqtrrid 3977	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
14	13	adantr 486	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
15		simpl 488	. . . . . 6 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})))
16	15	anim1i 627	. . . . 5 ⊢ (((((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})))
17		df-3an 1105	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) ↔ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
18	16, 17	sylibr 237	. . . 4 ⊢ (((((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})))
19		dif32 4251	. . . . . . . . . . 11 ⊢ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
20	3, 19	eqtri 2785	. . . . . . . . . 10 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
21	20	eleq2i 2854	. . . . . . . . 9 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}))
22		eldifsn 4751	. . . . . . . . 9 ⊢ (𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛 ≠ 𝑁))
23	21, 22	bitri 278	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛 ≠ 𝑁))
24	23	simplbi 502	. . . . . . 7 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝑉 ∖ {𝐾}))
25		eleq2 2851	. . . . . . 7 ⊢ ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑛 ∈ (𝑉 ∖ {𝐾})))
26	24, 25	imbitrrid 249	. . . . . 6 ⊢ ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
27	26	adantl 487	. . . . 5 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
28	27	imp 412	. . . 4 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾))
29	4, 5, 6, 7	nbupgrres 29810	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾)))
30	18, 28, 29	sylc 66	. . 3 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾))
31	14, 30	eqelssd 3955	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾}))
32	31	ex 418	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ∉ wnel 3063  {crab 3414   ∖ cdif 3899   ⊆ wss 3902  {csn 4587  {cpr 4589  ⟨cop 4593   I cid 5553   ↾ cres 5661  ‘cfv 6537  (class class class)co 7416  Vtxcvtx 29439  Edgcedg 29490  UPGraphcupgr 29523   NeighbVtx cnbgr 29778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-n0 12532  df-xnn0 12605  df-z 12619  df-uz 12891  df-fz 13564  df-hash 14397  df-vtx 29441  df-iedg 29442  df-edg 29491  df-upgr 29525  df-nbgr 29779

This theorem is used by:  uvtxupgrres  29854