Theorem frgrncvvdeq 28673

Description: In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 10-May-2021.)

Hypotheses

Ref	Expression
frgrncvvdeq.v	⊢ 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.d	⊢ 𝐷 = (VtxDeg‘𝐺)

Assertion

Ref	Expression
frgrncvvdeq	⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))

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

Allowed substitution hints:   𝐷(𝑥,𝑦)

Proof of Theorem frgrncvvdeq

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7310	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 NeighbVtx 𝑥) ∈ V)
2		frgrncvvdeq.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
3		eqid 2738	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4		eqid 2738	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5		eqid 2738	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6		simpl 483	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ∈ 𝑉)
7	6	ad2antlr 724	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ∈ 𝑉)
8		eldifi 4061	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦 ∈ 𝑉)
9	8	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦 ∈ 𝑉)
10	9	ad2antlr 724	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∈ 𝑉)
11		eldif 3897	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12		velsn 4577	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
13	12	biimpri 227	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 ∈ {𝑥})
14	13	equcoms 2023	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑦 ∈ {𝑥})
15	14	necon3bi 2970	. . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ {𝑥} → 𝑥 ≠ 𝑦)
16	11, 15	simplbiim 505	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥 ≠ 𝑦)
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
18	17	ad2antlr 724	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ≠ 𝑦)
19		simpr 485	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
20		simpl 483	. . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
21	20	adantr 481	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
22		eqid 2738	. . . . . . 7 ⊢ (𝑎 ∈ (𝐺 NeighbVtx 𝑥) ↦ (℩𝑏 ∈ (𝐺 NeighbVtx 𝑦){𝑎, 𝑏} ∈ (Edg‘𝐺))) = (𝑎 ∈ (𝐺 NeighbVtx 𝑥) ↦ (℩𝑏 ∈ (𝐺 NeighbVtx 𝑦){𝑎, 𝑏} ∈ (Edg‘𝐺)))
23	2, 3, 4, 5, 7, 10, 18, 19, 21, 22	frgrncvvdeqlem10 28672	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝑎 ∈ (𝐺 NeighbVtx 𝑥) ↦ (℩𝑏 ∈ (𝐺 NeighbVtx 𝑦){𝑎, 𝑏} ∈ (Edg‘𝐺))):(𝐺 NeighbVtx 𝑥)–1-1-onto→(𝐺 NeighbVtx 𝑦))
24	1, 23	hasheqf1od 14068	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑦)))
25		frgrusgr 28625	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	25, 6	anim12i 613	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉))
27	26	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉))
28	2	hashnbusgrvd 27895	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑥)) = ((VtxDeg‘𝐺)‘𝑥))
29	27, 28	syl 17	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = ((VtxDeg‘𝐺)‘𝑥))
30	25, 9	anim12i 613	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
31	30	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
32	2	hashnbusgrvd 27895	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑦)) = ((VtxDeg‘𝐺)‘𝑦))
33	31, 32	syl 17	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑦)) = ((VtxDeg‘𝐺)‘𝑦))
34	24, 29, 33	3eqtr3d 2786	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘𝑦))
35		frgrncvvdeq.d	. . . . 5 ⊢ 𝐷 = (VtxDeg‘𝐺)
36	35	fveq1i 6775	. . . 4 ⊢ (𝐷‘𝑥) = ((VtxDeg‘𝐺)‘𝑥)
37	35	fveq1i 6775	. . . 4 ⊢ (𝐷‘𝑦) = ((VtxDeg‘𝐺)‘𝑦)
38	34, 36, 37	3eqtr4g 2803	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷‘𝑥) = (𝐷‘𝑦))
39	38	ex 413	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
40	39	ralrimivva 3123	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  ∀wral 3064  Vcvv 3432   ∖ cdif 3884  {csn 4561  {cpr 4563   ↦ cmpt 5157  ‘cfv 6433  ℩crio 7231  (class class class)co 7275  ♯chash 14044  Vtxcvtx 27366  Edgcedg 27417  USGraphcusgr 27519   NeighbVtx cnbgr 27699  VtxDegcvtxdg 27832   FriendGraph cfrgr 28622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-xadd 12849  df-fz 13240  df-hash 14045  df-edg 27418  df-uhgr 27428  df-ushgr 27429  df-upgr 27452  df-umgr 27453  df-uspgr 27520  df-usgr 27521  df-nbgr 27700  df-vtxdg 27833  df-frgr 28623

This theorem is referenced by:  frgrwopreglem4a  28674