Theorem frgrncvvdeq 29541

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 7438	. . . . . 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 2733	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4		eqid 2733	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5		eqid 2733	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6		simpl 484	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ∈ 𝑉)
7	6	ad2antlr 726	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ∈ 𝑉)
8		eldifi 4124	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦 ∈ 𝑉)
9	8	adantl 483	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦 ∈ 𝑉)
10	9	ad2antlr 726	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∈ 𝑉)
11		eldif 3956	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12		velsn 4642	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
13	12	biimpri 227	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 ∈ {𝑥})
14	13	equcoms 2024	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑦 ∈ {𝑥})
15	14	necon3bi 2968	. . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ {𝑥} → 𝑥 ≠ 𝑦)
16	11, 15	simplbiim 506	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥 ≠ 𝑦)
17	16	adantl 483	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥 ≠ 𝑦)
18	17	ad2antlr 726	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥 ≠ 𝑦)
19		simpr 486	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
20		simpl 484	. . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
21	20	adantr 482	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
22		eqid 2733	. . . . . . 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 29540	. . . . . 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 14308	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑦)))
25		frgrusgr 29493	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	25, 6	anim12i 614	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉))
27	26	adantr 482	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥 ∈ 𝑉))
28	2	hashnbusgrvd 28764	. . . . . 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 614	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
31	30	adantr 482	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝑉))
32	2	hashnbusgrvd 28764	. . . . . 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 2781	. . . 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 6888	. . . 4 ⊢ (𝐷‘𝑥) = ((VtxDeg‘𝐺)‘𝑥)
37	35	fveq1i 6888	. . . 4 ⊢ (𝐷‘𝑦) = ((VtxDeg‘𝐺)‘𝑦)
38	34, 36, 37	3eqtr4g 2798	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷‘𝑥) = (𝐷‘𝑦))
39	38	ex 414	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))
40	39	ralrimivva 3201	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷‘𝑥) = (𝐷‘𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941   ∉ wnel 3047  ∀wral 3062  Vcvv 3475   ∖ cdif 3943  {csn 4626  {cpr 4628   ↦ cmpt 5229  ‘cfv 6539  ℩crio 7358  (class class class)co 7403  ♯chash 14285  Vtxcvtx 28235  Edgcedg 28286  USGraphcusgr 28388   NeighbVtx cnbgr 28568  VtxDegcvtxdg 28701   FriendGraph cfrgr 29490

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-1o 8460  df-2o 8461  df-oadd 8464  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-dju 9891  df-card 9929  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-nn 12208  df-2 12270  df-n0 12468  df-xnn0 12540  df-z 12554  df-uz 12818  df-xadd 13088  df-fz 13480  df-hash 14286  df-edg 28287  df-uhgr 28297  df-ushgr 28298  df-upgr 28321  df-umgr 28322  df-uspgr 28389  df-usgr 28390  df-nbgr 28569  df-vtxdg 28702  df-frgr 29491

This theorem is referenced by:  frgrwopreglem4a  29542