Theorem frgrncvvdeqlem3 30330

Description: Lemma 3 for frgrncvvdeq 30338. The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 12-Feb-2022.)

Hypotheses

Ref	Expression
frgrncvvdeq.v1	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e	⊢ 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx	⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny	⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrncvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrncvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrncvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrncvvdeq.f	⊢ (𝜑 → 𝐺 ∈ FriendGraph )
frgrncvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))

Assertion

Ref	Expression
frgrncvvdeqlem3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦   𝑦,𝑁

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥)   𝐺(𝑥)   𝑁(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem3

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrncvvdeq.ny	. . 3 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
2	1	ineq2i 4225	. 2 ⊢ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌))
3		frgrncvvdeq.f	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ FriendGraph )
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐺 ∈ FriendGraph )
5		frgrncvvdeq.nx	. . . . . . . 8 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
6	5	eleq2i 2831	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7		frgrncvvdeq.v1	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
8	7	nbgrisvtx 29373	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉))
10	6, 9	biimtrid 242	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝑉))
11	10	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝑉)
12		frgrncvvdeq.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑌 ∈ 𝑉)
14		frgrncvvdeq.xy	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
15		elnelne2 3056	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑌 ∉ 𝐷) → 𝑥 ≠ 𝑌)
16	14, 15	sylan2 593	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝜑) → 𝑥 ≠ 𝑌)
17	16	ancoms 458	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 𝑌)
18	11, 13, 17	3jca 1127	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
19		frgrncvvdeq.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
20	7, 19	frcond3 30298	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌) → ∃𝑛 ∈ 𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}))
21	4, 18, 20	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑛 ∈ 𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛})
22		vex 3482	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
23		elinsn 4715	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
24	22, 23	mpan 690	. . . . . . . . 9 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
25		frgrusgr 30290	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	19	nbusgreledg 29385	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27		prcom 4737	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑛, 𝑥} = {𝑥, 𝑛}
28	27	eleq1i 2830	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑛, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸)
29	26, 28	bitrdi 287	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, 𝑛} ∈ 𝐸))
30	29	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
31	3, 25, 30	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
32	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
33	32	com12 32	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, 𝑛} ∈ 𝐸))
34	33	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, 𝑛} ∈ 𝐸))
35	34	imp 406	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → {𝑥, 𝑛} ∈ 𝐸)
36	1	eqcomi 2744	. . . . . . . . . . . . . 14 ⊢ (𝐺 NeighbVtx 𝑌) = 𝑁
37	36	eleq2i 2831	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑛 ∈ 𝑁)
38	37	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛 ∈ 𝑁)
39	38	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑛 ∈ 𝑁)
40		frgrncvvdeq.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
41		frgrncvvdeq.ne	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
42		frgrncvvdeq.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
43	7, 19, 5, 1, 40, 12, 41, 14, 3, 42	frgrncvvdeqlem2 30329	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)
44		preq2 4739	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
45	44	eleq1d 2824	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
46	45	riota2 7413	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑁 ∧ ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
47	39, 43, 46	syl2an 596	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
48	35, 47	mpbid 232	. . . . . . . . 9 ⊢ (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
49	24, 48	sylan 580	. . . . . . . 8 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
50	49	eqcomd 2741	. . . . . . 7 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → 𝑛 = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
51	50	sneqd 4643	. . . . . 6 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → {𝑛} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
52		eqeq1 2739	. . . . . . 7 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
53	52	adantr 480	. . . . . 6 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} ↔ {𝑛} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
54	51, 53	mpbird 257	. . . . 5 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
55	54	ex 412	. . . 4 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
56	55	rexlimivw 3149	. . 3 ⊢ (∃𝑛 ∈ 𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
57	21, 56	mpcom 38	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
58	2, 57	eqtr2id 2788	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   ∉ wnel 3044  ∃wrex 3068  ∃!wreu 3376  Vcvv 3478   ∩ cin 3962  {csn 4631  {cpr 4633   ↦ cmpt 5231  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079  USGraphcusgr 29181   NeighbVtx cnbgr 29364   FriendGraph cfrgr 30287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-edg 29080  df-upgr 29114  df-umgr 29115  df-usgr 29183  df-nbgr 29365  df-frgr 30288

This theorem is referenced by:  frgrncvvdeqlem5  30332