Theorem frcond3 28661

Description: The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhood of a vertex and the neighborhood of a second, different vertex have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 30-Dec-2021.)

Hypotheses

Ref	Expression
frcond1.v	⊢ 𝑉 = (Vtx‘𝐺)
frcond1.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frcond3	⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐸   𝑥,𝐺   𝑥,𝑉

Proof of Theorem frcond3

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frcond1.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		frcond1.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	frcond1 28658	. . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
4	3	imp 406	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
5		ssrab2 4016	. . . . . . . . . 10 ⊢ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉
6		sseq1 3948	. . . . . . . . . 10 ⊢ ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
7	5, 6	mpbii 232	. . . . . . . . 9 ⊢ ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
8		vex 3438	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9	8	snss 4722	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑉 ↔ {𝑥} ⊆ 𝑉)
10	7, 9	sylibr 233	. . . . . . . 8 ⊢ ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → 𝑥 ∈ 𝑉)
11	10	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → 𝑥 ∈ 𝑉)
12		frgrusgr 28653	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
13	1, 2	nbusgr 27744	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸})
14	1, 2	nbusgr 27744	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐶) = {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸})
15	13, 14	ineq12d 4150	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USGraph → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
16	12, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ FriendGraph → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
18	17	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
19		inrab 4243	. . . . . . . . 9 ⊢ ({𝑏 ∈ 𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏 ∈ 𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}) = {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)}
20	18, 19	eqtrdi 2789	. . . . . . . 8 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)})
21		prcom 4671	. . . . . . . . . . . . . 14 ⊢ {𝐶, 𝑏} = {𝑏, 𝐶}
22	21	eleq1i 2824	. . . . . . . . . . . . 13 ⊢ ({𝐶, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝐶} ∈ 𝐸)
23	22	anbi2i 622	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
24		prex 5358	. . . . . . . . . . . . 13 ⊢ {𝐴, 𝑏} ∈ V
25		prex 5358	. . . . . . . . . . . . 13 ⊢ {𝑏, 𝐶} ∈ V
26	24, 25	prss 4756	. . . . . . . . . . . 12 ⊢ (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
27	23, 26	bitri 274	. . . . . . . . . . 11 ⊢ (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
28	27	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ 𝑏 ∈ 𝑉) → (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
29	28	rabbidva 3415	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏 ∈ 𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
31		simpr 484	. . . . . . . 8 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
32	20, 30, 31	3eqtrd 2777	. . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
33	11, 32	jca 511	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) ∧ {𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → (𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
34	33	ex 412	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ({𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → (𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
35	34	eximdv 1916	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → (∃𝑥{𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
36		reusn 4666	. . . 4 ⊢ (∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃𝑥{𝑏 ∈ 𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
37		df-rex 3069	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥} ↔ ∃𝑥(𝑥 ∈ 𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
38	35, 36, 37	3imtr4g 295	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → (∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
39	4, 38	mpd 15	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶)) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
40	39	ex 412	1 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1537  ∃wex 1777   ∈ wcel 2101   ≠ wne 2938  ∃wrex 3068  ∃!wreu 3219  {crab 3221   ∩ cin 3888   ⊆ wss 3889  {csn 4564  {cpr 4566  ‘cfv 6447  (class class class)co 7295  Vtxcvtx 27394  Edgcedg 27445  USGraphcusgr 27547   NeighbVtx cnbgr 27727   FriendGraph cfrgr 28650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608  ax-cnex 10955  ax-resscn 10956  ax-1cn 10957  ax-icn 10958  ax-addcl 10959  ax-addrcl 10960  ax-mulcl 10961  ax-mulrcl 10962  ax-mulcom 10963  ax-addass 10964  ax-mulass 10965  ax-distr 10966  ax-i2m1 10967  ax-1ne0 10968  ax-1rid 10969  ax-rnegex 10970  ax-rrecex 10971  ax-cnre 10972  ax-pre-lttri 10973  ax-pre-lttrn 10974  ax-pre-ltadd 10975  ax-pre-mulgt0 10976

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3908  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-int 4883  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-om 7733  df-1st 7851  df-2nd 7852  df-frecs 8117  df-wrecs 8148  df-recs 8222  df-rdg 8261  df-1o 8317  df-2o 8318  df-oadd 8321  df-er 8518  df-en 8754  df-dom 8755  df-sdom 8756  df-fin 8757  df-dju 9687  df-card 9725  df-pnf 11039  df-mnf 11040  df-xr 11041  df-ltxr 11042  df-le 11043  df-sub 11235  df-neg 11236  df-nn 12002  df-2 12064  df-n0 12262  df-xnn0 12334  df-z 12348  df-uz 12611  df-fz 13268  df-hash 14073  df-edg 27446  df-upgr 27480  df-umgr 27481  df-usgr 27549  df-nbgr 27728  df-frgr 28651

This theorem is referenced by:  frcond4  28662  frgrncvvdeqlem3  28693