Theorem frgrncvvdeqlem3 30359

Description: Lemma 3 for frgrncvvdeq 30367. 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 4148	. 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 2827	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7		frgrncvvdeq.v1	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
8	7	nbgrisvtx 29398	. . . . . . . 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 3046	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑌 ∉ 𝐷) → 𝑥 ≠ 𝑌)
16	14, 15	sylan2 594	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝜑) → 𝑥 ≠ 𝑌)
17	16	ancoms 458	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 𝑌)
18	11, 13, 17	3jca 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
19		frgrncvvdeq.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
20	7, 19	frcond3 30327	. . . 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 3431	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
23		elinsn 4644	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
24	22, 23	mpan 691	. . . . . . . . 9 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
25		frgrusgr 30319	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	19	nbusgreledg 29410	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27		prcom 4666	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑛, 𝑥} = {𝑥, 𝑛}
28	27	eleq1i 2826	. . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . 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 30358	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)
44		preq2 4668	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
45	44	eleq1d 2820	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
46	45	riota2 7338	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑁 ∧ ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
47	39, 43, 46	syl2an 597	. . . . . . . . . 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 581	. . . . . . . 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 4569	. . . . . 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 3132	. . 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 2783	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2930   ∉ wnel 3034  ∃wrex 3059  ∃!wreu 3338  Vcvv 3427   ∩ cin 3884  {csn 4557  {cpr 4559   ↦ cmpt 5155  ‘cfv 6487  ℩crio 7312  (class class class)co 7356  Vtxcvtx 29053  Edgcedg 29104  USGraphcusgr 29206   NeighbVtx cnbgr 29389   FriendGraph cfrgr 30316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-er 8632  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-dju 9814  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-n0 12427  df-xnn0 12500  df-z 12514  df-uz 12778  df-fz 13451  df-hash 14282  df-edg 29105  df-upgr 29139  df-umgr 29140  df-usgr 29208  df-nbgr 29390  df-frgr 30317

This theorem is referenced by:  frgrncvvdeqlem5  30361