Theorem frgrncvvdeq 30113

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 7450	. . . . . 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 2728	. . . . . . 7 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4		eqid 2728	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5		eqid 2728	. . . . . . 7 ⊢ (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6		simpl 482	. . . . . . . 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 4123	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦 ∈ 𝑉)
9	8	adantl 481	. . . . . . . 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 3955	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12		velsn 4641	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
13	12	biimpri 227	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → 𝑦 ∈ {𝑥})
14	13	equcoms 2016	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑦 ∈ {𝑥})
15	14	necon3bi 2963	. . . . . . . . . 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 726	. . . . . . 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 2728	. . . . . . 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 30112	. . . . . 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 14339	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑦)))
25		frgrusgr 30065	. . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
26	25, 6	anim12i 612	. . . . . . 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 29336	. . . . . 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 612	. . . . . . 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 29336	. . . . . 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 2776	. . . 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 6893	. . . 4 ⊢ (𝐷‘𝑥) = ((VtxDeg‘𝐺)‘𝑥)
37	35	fveq1i 6893	. . . 4 ⊢ (𝐷‘𝑦) = ((VtxDeg‘𝐺)‘𝑦)
38	34, 36, 37	3eqtr4g 2793	. . 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 3196	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 1534   ∈ wcel 2099   ≠ wne 2936   ∉ wnel 3042  ∀wral 3057  Vcvv 3470   ∖ cdif 3942  {csn 4625  {cpr 4627   ↦ cmpt 5226  ‘cfv 6543  ℩crio 7370  (class class class)co 7415  ♯chash 14316  Vtxcvtx 28803  Edgcedg 28854  USGraphcusgr 28956   NeighbVtx cnbgr 29139  VtxDegcvtxdg 29273   FriendGraph cfrgr 30062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-1o 8481  df-2o 8482  df-oadd 8485  df-er 8719  df-en 8959  df-dom 8960  df-sdom 8961  df-fin 8962  df-dju 9919  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-n0 12498  df-xnn0 12570  df-z 12584  df-uz 12848  df-xadd 13120  df-fz 13512  df-hash 14317  df-edg 28855  df-uhgr 28865  df-ushgr 28866  df-upgr 28889  df-umgr 28890  df-uspgr 28957  df-usgr 28958  df-nbgr 29140  df-vtxdg 29274  df-frgr 30063

This theorem is referenced by:  frgrwopreglem4a  30114