Theorem frgrncvvdeqlem3 30767

Description: Lemma 3 for frgrncvvdeq 30775. 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 4166	. 2 ⊢ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) = ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌))
3		frgrncvvdeq.f	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ FriendGraph )
4	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐺 ∈ FriendGraph )
5		frgrncvvdeq.nx	. . . . . . . 8 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
6	5	eleq2i 2854	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
7		frgrncvvdeq.v1	. . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺)
8	7	nbgrisvtx 29787	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉))
10	6, 9	biimtrid 245	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝑉))
11	10	imp 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝑉)
12		frgrncvvdeq.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
13	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑌 ∈ 𝑉)
14		frgrncvvdeq.xy	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
15		elnelne2 3075	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑌 ∉ 𝐷) → 𝑥 ≠ 𝑌)
16	14, 15	sylan2 605	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝜑) → 𝑥 ≠ 𝑌)
17	16	ancoms 464	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 𝑌)
18	11, 13, 17	3jca 1146	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
19		frgrncvvdeq.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
20	7, 19	frcond3 30735	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌) → ∃𝑛 ∈ 𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}))
21	4, 18, 20	sylc 66	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑛 ∈ 𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛})
22		vex 3457	. . . . . . . . . 10 ⊢ 𝑛 ∈ V
23		elinsn 4674	. . . . . . . . . 10 ⊢ ((𝑛 ∈ V ∧ ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛}) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
24	22, 23	mpan 703	. . . . . . . . 9 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)))
25		frgrusgr 30727	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	19	nbusgreledg 29799	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑛, 𝑥} ∈ 𝐸))
27		prcom 4696	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑛, 𝑥} = {𝑥, 𝑛}
28	27	eleq1i 2853	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑛, 𝑥} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸)
29	26, 28	bitrdi 290	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) ↔ {𝑥, 𝑛} ∈ 𝐸))
30	29	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
31	3, 25, 30	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
32	31	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → {𝑥, 𝑛} ∈ 𝐸))
33	32	com12 33	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑥) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, 𝑛} ∈ 𝐸))
34	33	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, 𝑛} ∈ 𝐸))
35	34	imp 412	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → {𝑥, 𝑛} ∈ 𝐸)
36	1	eqcomi 2771	. . . . . . . . . . . . 13 ⊢ (𝐺 NeighbVtx 𝑌) = 𝑁
37	36	eleq2i 2854	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑛 ∈ 𝑁)
38	37	bilani 510	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑛 ∈ 𝑁)
39		frgrncvvdeq.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
40		frgrncvvdeq.ne	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
41		frgrncvvdeq.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
42	7, 19, 5, 1, 39, 12, 40, 14, 3, 41	frgrncvvdeqlem2 30766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)
43		preq2 4698	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑛 → {𝑥, 𝑦} = {𝑥, 𝑛})
44	43	eleq1d 2847	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑛 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑥, 𝑛} ∈ 𝐸))
45	44	riota2 7398	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑁 ∧ ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
46	38, 42, 45	syl2an 608	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → ({𝑥, 𝑛} ∈ 𝐸 ↔ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛))
47	35, 46	mpbid 235	. . . . . . . . 9 ⊢ (((𝑛 ∈ (𝐺 NeighbVtx 𝑥) ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
48	24, 47	sylan 592	. . . . . . . 8 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = 𝑛)
49	48	eqcomd 2768	. . . . . . 7 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → 𝑛 = (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
50	49	sneqd 4599	. . . . . 6 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → {𝑛} = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
51		eqeq1 2766	. . . . . . 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)}))
52	51	adantr 486	. . . . . 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)}))
53	50, 52	mpbird 260	. . . . 5 ⊢ ((((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} ∧ (𝜑 ∧ 𝑥 ∈ 𝐷)) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
54	53	ex 418	. . . 4 ⊢ (((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
55	54	rexlimivw 3161	. . 3 ⊢ (∃𝑛 ∈ 𝑉 ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {𝑛} → ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)}))
56	21, 55	mpcom 39	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 NeighbVtx 𝑥) ∩ (𝐺 NeighbVtx 𝑌)) = {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)})
57	2, 56	eqtr2id 2810	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   ∉ wnel 3063  ∃wrex 3088  ∃!wreu 3365  Vcvv 3453   ∩ cin 3901  {csn 4587  {cpr 4589   ↦ cmpt 5190  ‘cfv 6537  ℩crio 7372  (class class class)co 7416  Vtxcvtx 29439  Edgcedg 29490  USGraphcusgr 29595   NeighbVtx cnbgr 29778   FriendGraph cfrgr 30724

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-rmo 3367  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-edg 29491  df-upgr 29525  df-umgr 29526  df-usgr 29597  df-nbgr 29779  df-frgr 30725

This theorem is used by:  frgrncvvdeqlem5  30769