Theorem frisusgranb 26524

Description: In a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.)

Assertion

Ref	Expression
frisusgranb	⊢ (𝑉 FriendGrph 𝐸 → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥})

Distinct variable groups:   𝑘,𝑉,𝑙,𝑥   𝑘,𝐸,𝑙,𝑥

Proof of Theorem frisusgranb

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frisusgrapr 26518	. 2 ⊢ (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛 ∈ 𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸))
2		ssrab2 3650	. . . . . . . . . . . 12 ⊢ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} ⊆ 𝑉
3		sseq1 3589	. . . . . . . . . . . 12 ⊢ ({𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥} → ({𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
4	2, 3	mpbii 222	. . . . . . . . . . 11 ⊢ ({𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
5		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
6	5	snss 4259	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑉 ↔ {𝑥} ⊆ 𝑉)
7	4, 6	sylibr 223	. . . . . . . . . 10 ⊢ ({𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥} → 𝑥 ∈ 𝑉)
8	7	adantl 481	. . . . . . . . 9 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥}) → 𝑥 ∈ 𝑉)
9		nbusgra 25957	. . . . . . . . . . . . 13 ⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑘) = {𝑛 ∈ 𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸})
10		nbusgra 25957	. . . . . . . . . . . . 13 ⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑙) = {𝑛 ∈ 𝑉 ∣ {𝑙, 𝑛} ∈ ran 𝐸})
11	9, 10	ineq12d 3777	. . . . . . . . . . . 12 ⊢ (𝑉 USGrph 𝐸 → ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = ({𝑛 ∈ 𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸} ∩ {𝑛 ∈ 𝑉 ∣ {𝑙, 𝑛} ∈ ran 𝐸}))
12	11	ad3antrrr 762	. . . . . . . . . . 11 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥}) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = ({𝑛 ∈ 𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸} ∩ {𝑛 ∈ 𝑉 ∣ {𝑙, 𝑛} ∈ ran 𝐸}))
13		inrab 3858	. . . . . . . . . . 11 ⊢ ({𝑛 ∈ 𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸} ∩ {𝑛 ∈ 𝑉 ∣ {𝑙, 𝑛} ∈ ran 𝐸}) = {𝑛 ∈ 𝑉 ∣ ({𝑘, 𝑛} ∈ ran 𝐸 ∧ {𝑙, 𝑛} ∈ ran 𝐸)}
14	12, 13	syl6eq 2660	. . . . . . . . . 10 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥}) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑛 ∈ 𝑉 ∣ ({𝑘, 𝑛} ∈ ran 𝐸 ∧ {𝑙, 𝑛} ∈ ran 𝐸)})
15		prcom 4211	. . . . . . . . . . . . . . . 16 ⊢ {𝑘, 𝑛} = {𝑛, 𝑘}
16	15	eleq1i 2679	. . . . . . . . . . . . . . 15 ⊢ ({𝑘, 𝑛} ∈ ran 𝐸 ↔ {𝑛, 𝑘} ∈ ran 𝐸)
17		prcom 4211	. . . . . . . . . . . . . . . 16 ⊢ {𝑙, 𝑛} = {𝑛, 𝑙}
18	17	eleq1i 2679	. . . . . . . . . . . . . . 15 ⊢ ({𝑙, 𝑛} ∈ ran 𝐸 ↔ {𝑛, 𝑙} ∈ ran 𝐸)
19	16, 18	anbi12i 729	. . . . . . . . . . . . . 14 ⊢ (({𝑘, 𝑛} ∈ ran 𝐸 ∧ {𝑙, 𝑛} ∈ ran 𝐸) ↔ ({𝑛, 𝑘} ∈ ran 𝐸 ∧ {𝑛, 𝑙} ∈ ran 𝐸))
20		zfpair2 4834	. . . . . . . . . . . . . . 15 ⊢ {𝑛, 𝑘} ∈ V
21		zfpair2 4834	. . . . . . . . . . . . . . 15 ⊢ {𝑛, 𝑙} ∈ V
22	20, 21	prss 4291	. . . . . . . . . . . . . 14 ⊢ (({𝑛, 𝑘} ∈ ran 𝐸 ∧ {𝑛, 𝑙} ∈ ran 𝐸) ↔ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸)
23	19, 22	bitri 263	. . . . . . . . . . . . 13 ⊢ (({𝑘, 𝑛} ∈ ran 𝐸 ∧ {𝑙, 𝑛} ∈ ran 𝐸) ↔ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸)
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ 𝑛 ∈ 𝑉) → (({𝑘, 𝑛} ∈ ran 𝐸 ∧ {𝑙, 𝑛} ∈ ran 𝐸) ↔ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸))
25	24	rabbidva 3163	. . . . . . . . . . 11 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → {𝑛 ∈ 𝑉 ∣ ({𝑘, 𝑛} ∈ ran 𝐸 ∧ {𝑙, 𝑛} ∈ ran 𝐸)} = {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸})
26	25	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥}) → {𝑛 ∈ 𝑉 ∣ ({𝑘, 𝑛} ∈ ran 𝐸 ∧ {𝑙, 𝑛} ∈ ran 𝐸)} = {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸})
27		simpr 476	. . . . . . . . . 10 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥}) → {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥})
28	14, 26, 27	3eqtrd 2648	. . . . . . . . 9 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥}) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥})
29	8, 28	jca 553	. . . . . . . 8 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) ∧ {𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥}) → (𝑥 ∈ 𝑉 ∧ ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥}))
30	29	ex 449	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → ({𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥} → (𝑥 ∈ 𝑉 ∧ ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥})))
31	30	eximdv 1833	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → (∃𝑥{𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥} → ∃𝑥(𝑥 ∈ 𝑉 ∧ ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥})))
32		reusn 4206	. . . . . 6 ⊢ (∃!𝑛 ∈ 𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑥{𝑛 ∈ 𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸} = {𝑥})
33		df-rex 2902	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑉 ∧ ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥}))
34	31, 32, 33	3imtr4g 284	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → (∃!𝑛 ∈ 𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸 → ∃𝑥 ∈ 𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥}))
35	34	ralimdva 2945	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑘 ∈ 𝑉) → (∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛 ∈ 𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥}))
36	35	ralimdva 2945	. . 3 ⊢ (𝑉 USGrph 𝐸 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛 ∈ 𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸 → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥}))
37	36	imp 444	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛 ∈ 𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ ran 𝐸) → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥})
38	1, 37	syl 17	1 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946   FriendGrph cfrgra 26515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-frgra 26516

This theorem is referenced by:  frgrancvvdeqlem4  26560