Theorem frgrncvvdeq 30257

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 7384	. . . . . 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 2729	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4		eqid 2729	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5		eqid 2729	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ∈ 𝑉)
7	6	ad2antlr 727	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ∈ 𝑉)
8		eldifi 4082	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦 ∈ 𝑉)
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦 ∈ 𝑉)
10	9	ad2antlr 727	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∈ 𝑉)
11		eldif 3913	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12		velsn 4593	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
13	12	biimpri 228	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 ∈ {𝑥})
14	13	equcoms 2020	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑦 ∈ {𝑥})
15	14	necon3bi 2951	. . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ {𝑥} → 𝑥 ≠ 𝑦)
16	11, 15	simplbiim 504	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥 ≠ 𝑦)
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
18	17	ad2antlr 727	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ≠ 𝑦)
19		simpr 484	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
20		simpl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
21	20	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
22		eqid 2729	. . . . . . 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 30256	. . . . . 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 14260	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑦)))
25		frgrusgr 30209	. . . . . . . 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 480	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉))
28	2	hashnbusgrvd 29478	. . . . . 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 480	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
32	2	hashnbusgrvd 29478	. . . . . 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 2772	. . . 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 6823	. . . 4 ⊢ (𝐷‘𝑥) = ((VtxDeg‘𝐺)‘𝑥)
37	35	fveq1i 6823	. . . 4 ⊢ (𝐷‘𝑦) = ((VtxDeg‘𝐺)‘𝑦)
38	34, 36, 37	3eqtr4g 2789	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷‘𝑥) = (𝐷‘𝑦))
39	38	ex 412	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
40	39	ralrimivva 3172	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∉ wnel 3029  ∀wral 3044  Vcvv 3436   ∖ cdif 3900  {csn 4577  {cpr 4579   ↦ cmpt 5173  ‘cfv 6482  ℩crio 7305  (class class class)co 7349  ♯chash 14237  Vtxcvtx 28945  Edgcedg 28996  USGraphcusgr 29098   NeighbVtx cnbgr 29281  VtxDegcvtxdg 29415   FriendGraph cfrgr 30206

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-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-xadd 13015  df-fz 13411  df-hash 14238  df-edg 28997  df-uhgr 29007  df-ushgr 29008  df-upgr 29031  df-umgr 29032  df-uspgr 29099  df-usgr 29100  df-nbgr 29282  df-vtxdg 29416  df-frgr 30207

This theorem is referenced by:  frgrwopreglem4a  30258