Theorem frgrncvvdeqlem2 30391

Description: Lemma 2 for frgrncvvdeq 30400. 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐺 ∈ FriendGraph )
3		frgrncvvdeq.nx	. . . . . . 7 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋)
4	3	eleq2i 2829	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5		frgrncvvdeq.v1	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	nbgrisvtx 29430	. . . . . . 7 ⊢ (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉)
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉))
8	4, 7	biimtrid 242	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝑉))
9	8	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝑉)
10		frgrncvvdeq.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑌 ∈ 𝑉)
12		frgrncvvdeq.xy	. . . . . 6 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
13		elnelne2 3049	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑌 ∉ 𝐷) → 𝑥 ≠ 𝑌)
14	13	expcom 413	. . . . . 6 ⊢ (𝑌 ∉ 𝐷 → (𝑥 ∈ 𝐷 → 𝑥 ≠ 𝑌))
15	12, 14	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → 𝑥 ≠ 𝑌))
16	15	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ≠ 𝑌)
17	9, 11, 16	3jca 1129	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
18		frgrncvvdeq.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
19	5, 18	frcond1 30357	. . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌) → ∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
20	2, 17, 19	sylc 65	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
21		frgrusgr 30352	. . . 4 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22		prex 5379	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
23		prex 5379	. . . . . . . . . . . 12 ⊢ {𝑦, 𝑌} ∈ V
24	22, 23	prss 4764	. . . . . . . . . . 11 ⊢ (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸)
25		ancom 460	. . . . . . . . . . 11 ⊢ (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑌} ∈ 𝐸) ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
26	24, 25	bitr3i 277	. . . . . . . . . 10 ⊢ ({{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))
27	26	anbi2i 624	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦 ∈ 𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
28		usgrumgr 29270	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
29	5, 18	umgrpredgv 29229	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
30	29	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦 ∈ 𝑉)
31	30	ex 412	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ UMGraph → ({𝑥, 𝑦} ∈ 𝐸 → 𝑦 ∈ 𝑉))
32	28, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → ({𝑥, 𝑦} ∈ 𝐸 → 𝑦 ∈ 𝑉))
33	32	adantld 490	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) → 𝑦 ∈ 𝑉))
34	33	pm4.71rd 562	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦 ∈ 𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸))))
35	27, 34	bitr4id 290	. . . . . . . 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 2829	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑁 ↔ 𝑦 ∈ (𝐺 NeighbVtx 𝑌))
38	18	nbusgreledg 29442	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑦, 𝑌} ∈ 𝐸))
39	37, 38	bitr2id 284	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → ({𝑦, 𝑌} ∈ 𝐸 ↔ 𝑦 ∈ 𝑁))
40	39	anbi1d 632	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸) ↔ (𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
41	35, 40	bitrd 279	. . . . . . 7 ⊢ (𝐺 ∈ USGraph → ((𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
42	41	eubidv 2587	. . . . . 6 ⊢ (𝐺 ∈ USGraph → (∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ ∃!𝑦(𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
43	42	biimpd 229	. . . . 5 ⊢ (𝐺 ∈ USGraph → (∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
44		df-reu 3344	. . . . 5 ⊢ (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45		df-reu 3344	. . . . 5 ⊢ (∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸 ↔ ∃!𝑦(𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸))
46	43, 44, 45	3imtr4g 296	. . . 4 ⊢ (𝐺 ∈ USGraph → (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
47	1, 21, 46	3syl 18	. . 3 ⊢ (𝜑 → (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
48	47	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸))
49	20, 48	mpd 15	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃!weu 2569   ≠ wne 2933   ∉ wnel 3037  ∃!wreu 3341   ⊆ wss 3890  {cpr 4570   ↦ cmpt 5167  ‘cfv 6496  ℩crio 7320  (class class class)co 7364  Vtxcvtx 29085  Edgcedg 29136  UMGraphcumgr 29170  USGraphcusgr 29238   NeighbVtx cnbgr 29421   FriendGraph cfrgr 30349

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 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112

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 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-1st 7939  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9822  df-card 9860  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-nn 12172  df-2 12241  df-n0 12435  df-xnn0 12508  df-z 12522  df-uz 12786  df-fz 13459  df-hash 14290  df-edg 29137  df-upgr 29171  df-umgr 29172  df-usgr 29240  df-nbgr 29422  df-frgr 30350

This theorem is referenced by:  frgrncvvdeqlem3  30392  frgrncvvdeqlem4  30393