Theorem frgrncvvdeqlem2 30279

Description: Lemma 2 for frgrncvvdeq 30288. 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 2820	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5		frgrncvvdeq.v1	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	nbgrisvtx 29321	. . . . . . 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 3041	. . . . . . 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 1128	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
18		frgrncvvdeq.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
19	5, 18	frcond1 30245	. . 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 30240	. . . 4 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22		prex 5387	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
23		prex 5387	. . . . . . . . . . . 12 ⊢ {𝑦, 𝑌} ∈ V
24	22, 23	prss 4780	. . . . . . . . . . 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 623	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦 ∈ 𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
28		usgrumgr 29161	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
29	5, 18	umgrpredgv 29120	. . . . . . . . . . . . . 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 2820	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑁 ↔ 𝑦 ∈ (𝐺 NeighbVtx 𝑌))
38	18	nbusgreledg 29333	. . . . . . . . . 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 631	. . . . . . . 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 2579	. . . . . 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 3352	. . . . 5 ⊢ (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45		df-reu 3352	. . . . 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 1086   = wceq 1540   ∈ wcel 2109  ∃!weu 2561   ≠ wne 2925   ∉ wnel 3029  ∃!wreu 3349   ⊆ wss 3911  {cpr 4587   ↦ cmpt 5183  ‘cfv 6499  ℩crio 7325  (class class class)co 7369  Vtxcvtx 28976  Edgcedg 29027  UMGraphcumgr 29061  USGraphcusgr 29129   NeighbVtx cnbgr 29312   FriendGraph cfrgr 30237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-hash 14272  df-edg 29028  df-upgr 29062  df-umgr 29063  df-usgr 29131  df-nbgr 29313  df-frgr 30238

This theorem is referenced by:  frgrncvvdeqlem3  30280  frgrncvvdeqlem4  30281