Theorem frgrncvvdeqlem2 30928

Description: Lemma 2 for frgrncvvdeq 30937. In a friendship graph, for each neighbor of a vertex there is exactly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-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
frgrncvvdeqlem2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)

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

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

Proof of Theorem frgrncvvdeqlem2

Step	Hyp	Ref	Expression
1		frgrncvvdeq.f	. . . 4 ⊢ (𝜑 → 𝐺 ∈ FriendGraph )
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐺 ∈ FriendGraph )
3		frgrncvvdeq.nx	. . . . . . 7 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
4	3	eleq2i 2856	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5		frgrncvvdeq.v1	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	nbgrisvtx 29949	. . . . . . 7 ⊢ (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉)
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉))
8	4, 7	biimtrid 245	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝑉))
9	8	imp 412	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝑉)
10		frgrncvvdeq.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑌 ∈ 𝑉)
12		frgrncvvdeq.xy	. . . . . 6 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
13		elnelne2 3077	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑌 ∉ 𝐷) → 𝑥 ≠ 𝑌)
14	13	expcom 419	. . . . . 6 ⊢ (𝑌 ∉ 𝐷 → (𝑥 ∈ 𝐷 → 𝑥 ≠ 𝑌))
15	12, 14	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → 𝑥 ≠ 𝑌))
16	15	imp 412	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 𝑌)
17	9, 11, 16	3jca 1146	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
18		frgrncvvdeq.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
19	5, 18	frcond1 30894	. . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌) → ∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
20	2, 17, 19	sylc 66	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
21		frgrusgr 30889	. . . 4 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22		prex 5400	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
23		prex 5400	. . . . . . . . . . . 12 ⊢ {𝑦, 𝑌} ∈ V
24	22, 23	prss 4779	. . . . . . . . . . 11 ⊢ (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
25		ancom 466	. . . . . . . . . . 11 ⊢ (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
26	24, 25	bitr3i 280	. . . . . . . . . 10 ⊢ ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
27	26	anbi2i 635	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦 ∈ 𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
28		usgrumgr 29789	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
29	5, 18	umgrpredgv 29745	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
30	29	simprd 501	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦 ∈ 𝑉)
31	30	ex 418	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UMGraph → ({𝑥, 𝑦} ∈ 𝐸 → 𝑦 ∈ 𝑉))
32	28, 31	syl 18	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → ({𝑥, 𝑦} ∈ 𝐸 → 𝑦 ∈ 𝑉))
33	32	adantld 496	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦 ∈ 𝑉))
34	33	pm4.71rd 572	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦 ∈ 𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))))
35	27, 34	bitr4id 293	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ((𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
36		frgrncvvdeq.ny	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌)
37	36	eleq2i 2856	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑁 ↔ 𝑦 ∈ (𝐺 NeighbVtx 𝑌))
38	18	nbusgreledg 29961	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑦, 𝑌} ∈ 𝐸))
39	37, 38	bitr2id 287	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → ({𝑦, 𝑌} ∈ 𝐸 ↔ 𝑦 ∈ 𝑁))
40	39	anbi1d 643	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
41	35, 40	bitrd 282	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ((𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
42	41	eubidv 2614	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ∃!𝑦(𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
43	42	biimpd 232	. . . . 5 ⊢ (𝐺 ∈ USGraph → (∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
44		df-reu 3370	. . . . 5 ⊢ (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45		df-reu 3370	. . . . 5 ⊢ (∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸 ↔ ∃!𝑦(𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
46	43, 44, 45	3imtr4g 299	. . . 4 ⊢ (𝐺 ∈ USGraph → (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
47	1, 21, 46	3syl 19	. . 3 ⊢ (𝜑 → (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
48	47	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
49	20, 48	mpd 16	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃!weu 2596   ≠ wne 2959   ∉ wnel 3065  ∃!wreu 3367   ⊆ wss 3900  {cpr 4590   ↦ cmpt 5189  ‘cfv 6555  ℩crio 7396  (class class class)co 7440  Vtxcvtx 29601  Edgcedg 29652  UMGraphcumgr 29686  USGraphcusgr 29757   NeighbVtx cnbgr 29940   FriendGraph cfrgr 30886

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-n0 12631  df-xnn0 12704  df-z 12718  df-uz 12990  df-fz 13664  df-hash 14499  df-edg 29653  df-upgr 29687  df-umgr 29688  df-usgr 29759  df-nbgr 29941  df-frgr 30887

This theorem is used by:  frgrncvvdeqlem3  30929  frgrncvvdeqlem4  30930