Theorem frgrncvvdeqlem3 30380

Description: Lemma 3 for frgrncvvdeq 30388. 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 4170	. 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 2829	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7		frgrncvvdeq.v1	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
8	7	nbgrisvtx 29418	. . . . . . . 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 3049	. . . . . . 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 30348	. . . 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 3445	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
23		elinsn 4668	. . . . . . . . . 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 30340	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	19	nbusgreledg 29430	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27		prcom 4690	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑛, 𝑥} = {𝑥, 𝑛}
28	27	eleq1i 2828	. . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . 14 ⊢ (𝐺 NeighbVtx 𝑌) = 𝑁
37	36	eleq2i 2829	. . . . . . . . . . . . 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 30379	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)
44		preq2 4692	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
45	44	eleq1d 2822	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
46	45	riota2 7342	. . . . . . . . . . 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 2743	. . . . . . 7 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → 𝑛 = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
51	50	sneqd 4593	. . . . . 6 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → {𝑛} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
52		eqeq1 2741	. . . . . . 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 3134	. . 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 2785	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 2933   ∉ wnel 3037  ∃wrex 3061  ∃!wreu 3349  Vcvv 3441   ∩ cin 3901  {csn 4581  {cpr 4583   ↦ cmpt 5180  ‘cfv 6493  ℩crio 7316  (class class class)co 7360  Vtxcvtx 29073  Edgcedg 29124  USGraphcusgr 29226   NeighbVtx cnbgr 29409   FriendGraph cfrgr 30337

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 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-n0 12406  df-xnn0 12479  df-z 12493  df-uz 12756  df-fz 13428  df-hash 14258  df-edg 29125  df-upgr 29159  df-umgr 29160  df-usgr 29228  df-nbgr 29410  df-frgr 30338

This theorem is referenced by:  frgrncvvdeqlem5  30382