Theorem frgrncvvdeqlem3 28566

Description: Lemma 3 for frgrncvvdeq 28574. 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 4140	. 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 2830	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7		frgrncvvdeq.v1	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
8	7	nbgrisvtx 27611	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉))
10	6, 9	syl5bi 241	. . . . . 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 3059	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑌 ∉ 𝐷) → 𝑥 ≠ 𝑌)
16	14, 15	sylan2 592	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝜑) → 𝑥 ≠ 𝑌)
17	16	ancoms 458	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 𝑌)
18	11, 13, 17	3jca 1126	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
19		frgrncvvdeq.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
20	7, 19	frcond3 28534	. . . 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 3426	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
23		elinsn 4643	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
24	22, 23	mpan 686	. . . . . . . . 9 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
25		frgrusgr 28526	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	19	nbusgreledg 27623	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27		prcom 4665	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑛, 𝑥} = {𝑥, 𝑛}
28	27	eleq1i 2829	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑛, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸)
29	26, 28	bitrdi 286	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, 𝑛} ∈ 𝐸))
30	29	biimpd 228	. . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . 14 ⊢ (𝐺 NeighbVtx 𝑌) = 𝑁
37	36	eleq2i 2830	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑛 ∈ 𝑁)
38	37	biimpi 215	. . . . . . . . . . . 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 28565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)
44		preq2 4667	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
45	44	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
46	45	riota2 7238	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑁 ∧ ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
47	39, 43, 46	syl2an 595	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
49	24, 48	sylan 579	. . . . . . . 8 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
50	49	eqcomd 2744	. . . . . . 7 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → 𝑛 = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
51	50	sneqd 4570	. . . . . 6 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → {𝑛} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
52		eqeq1 2742	. . . . . . 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 256	. . . . 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 3210	. . 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 2792	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∉ wnel 3048  ∃wrex 3064  ∃!wreu 3065  Vcvv 3422   ∩ cin 3882  {csn 4558  {cpr 4560   ↦ cmpt 5153  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Vtxcvtx 27269  Edgcedg 27320  USGraphcusgr 27422   NeighbVtx cnbgr 27602   FriendGraph cfrgr 28523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-edg 27321  df-upgr 27355  df-umgr 27356  df-usgr 27424  df-nbgr 27603  df-frgr 28524

This theorem is referenced by:  frgrncvvdeqlem5  28568