Theorem frgrncvvdeqlem2 29821

Description: Lemma 2 for frgrncvvdeq 29830. 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 2824	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ (𝐺 NeighbVtx 𝑋))
5		frgrncvvdeq.v1	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	nbgrisvtx 28866	. . . . . . 7 ⊢ (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉)
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐺 NeighbVtx 𝑋) → 𝑥 ∈ 𝑉))
8	4, 7	biimtrid 241	. . . . 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 3057	. . . . . . 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 1127	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ≠ 𝑌))
18		frgrncvvdeq.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
19	5, 18	frcond1 29787	. . 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 29782	. . . 4 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
22		prex 5432	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦} ∈ V
23		prex 5432	. . . . . . . . . . . 12 ⊢ {𝑦, 𝑌} ∈ V
24	22, 23	prss 4823	. . . . . . . . . . 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 622	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) ↔ (𝑦 ∈ 𝑉 ∧ ({𝑦, 𝑌} ∈ 𝐸 ∧ {𝑥, 𝑦} ∈ 𝐸)))
28		usgrumgr 28707	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
29	5, 18	umgrpredgv 28668	. . . . . . . . . . . . . 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 2824	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑁 ↔ 𝑦 ∈ (𝐺 NeighbVtx 𝑌))
38	18	nbusgreledg 28878	. . . . . . . . . 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 629	. . . . . . . 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 228	. . . . 5 ⊢ (𝐺 ∈ USGraph → (∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸) → ∃!𝑦(𝑦 ∈ 𝑁 ∧ {𝑥, 𝑦} ∈ 𝐸)))
44		df-reu 3376	. . . . 5 ⊢ (∃!𝑦 ∈ 𝑉 {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸 ↔ ∃!𝑦(𝑦 ∈ 𝑉 ∧ {{𝑥, 𝑦}, {𝑦, 𝑌}} ⊆ 𝐸))
45		df-reu 3376	. . . . 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 2105  ∃!weu 2561   ≠ wne 2939   ∉ wnel 3045  ∃!wreu 3373   ⊆ wss 3948  {cpr 4630   ↦ cmpt 5231  ‘cfv 6543  ℩crio 7367  (class class class)co 7412  Vtxcvtx 28524  Edgcedg 28575  UMGraphcumgr 28609  USGraphcusgr 28677   NeighbVtx cnbgr 28857   FriendGraph cfrgr 29779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-hash 14296  df-edg 28576  df-upgr 28610  df-umgr 28611  df-usgr 28679  df-nbgr 28858  df-frgr 29780

This theorem is referenced by:  frgrncvvdeqlem3  29822  frgrncvvdeqlem4  29823