Theorem frgrncvvdeq 30513

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 7433	. . . . . 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 2764	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4		eqid 2764	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5		eqid 2764	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6		simpl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ∈ 𝑉)
7	6	ad2antlr 737	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ∈ 𝑉)
8		eldifi 4086	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦 ∈ 𝑉)
9	8	adantl 485	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦 ∈ 𝑉)
10	9	ad2antlr 737	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∈ 𝑉)
11		eldif 3916	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12		velsn 4600	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
13	12	biimpri 230	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 ∈ {𝑥})
14	13	equcoms 2042	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑦 ∈ {𝑥})
15	14	necon3bi 2985	. . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ {𝑥} → 𝑥 ≠ 𝑦)
16	11, 15	simplbiim 512	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥 ≠ 𝑦)
17	16	adantl 485	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
18	17	ad2antlr 737	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ≠ 𝑦)
19		simpr 488	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
20		simpl 486	. . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
21	20	adantr 484	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
22		eqid 2764	. . . . . . 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 30512	. . . . . 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 14368	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑦)))
25		frgrusgr 30465	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	25, 6	anim12i 622	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉))
27	26	adantr 484	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉))
28	2	hashnbusgrvd 29731	. . . . . 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 622	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
31	30	adantr 484	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
32	2	hashnbusgrvd 29731	. . . . . 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 2807	. . . 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 6870	. . . 4 ⊢ (𝐷‘𝑥) = ((VtxDeg‘𝐺)‘𝑥)
37	35	fveq1i 6870	. . . 4 ⊢ (𝐷‘𝑦) = ((VtxDeg‘𝐺)‘𝑦)
38	34, 36, 37	3eqtr4g 2824	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷‘𝑥) = (𝐷‘𝑦))
39	38	ex 416	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
40	39	ralrimivva 3207	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144   ≠ wne 2959   ∉ wnel 3063  ∀wral 3078  Vcvv 3456   ∖ cdif 3903  {csn 4584  {cpr 4586   ↦ cmpt 5183  ‘cfv 6523  ℩crio 7354  (class class class)co 7398  ♯chash 14345  Vtxcvtx 29199  Edgcedg 29250  USGraphcusgr 29352   NeighbVtx cnbgr 29535  VtxDegcvtxdg 29668   FriendGraph cfrgr 30462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-n0 12484  df-xnn0 12557  df-z 12571  df-uz 12842  df-xadd 13117  df-fz 13515  df-hash 14346  df-edg 29251  df-uhgr 29261  df-ushgr 29262  df-upgr 29285  df-umgr 29286  df-uspgr 29353  df-usgr 29354  df-nbgr 29536  df-vtxdg 29669  df-frgr 30463

This theorem is referenced by:  frgrwopreglem4a  30514