MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frcond1 Structured version   Visualization version   GIF version

Theorem frcond1 27615
Description: The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
frcond1.v 𝑉 = (Vtx‘𝐺)
frcond1.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frcond1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑉,𝑏
Allowed substitution hint:   𝐸(𝑏)

Proof of Theorem frcond1
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frcond1.v . . 3 𝑉 = (Vtx‘𝐺)
2 frcond1.e . . 3 𝐸 = (Edg‘𝐺)
31, 2frgrusgrfrcond 27608 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4 preq2 4458 . . . . . . 7 (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
54preq1d 4463 . . . . . 6 (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
65sseq1d 3828 . . . . 5 (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
76reubidv 3309 . . . 4 (𝑘 = 𝐴 → (∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8 preq2 4458 . . . . . . 7 (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
98preq2d 4464 . . . . . 6 (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
109sseq1d 3828 . . . . 5 (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
1110reubidv 3309 . . . 4 (𝑙 = 𝐶 → (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12 simp1 1167 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐴𝑉)
13 sneq 4378 . . . . . 6 (𝑘 = 𝐴 → {𝑘} = {𝐴})
1413difeq2d 3926 . . . . 5 (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
1514adantl 474 . . . 4 (((𝐴𝑉𝐶𝑉𝐴𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16 necom 3024 . . . . . . . 8 (𝐴𝐶𝐶𝐴)
1716biimpi 208 . . . . . . 7 (𝐴𝐶𝐶𝐴)
1817anim2i 611 . . . . . 6 ((𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
19183adant1 1161 . . . . 5 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
20 eldifsn 4506 . . . . 5 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉𝐶𝐴))
2119, 20sylibr 226 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
227, 11, 12, 15, 21rspc2vd 27614 . . 3 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
23 prcom 4456 . . . . . . 7 {𝑏, 𝐴} = {𝐴, 𝑏}
2423preq1i 4460 . . . . . 6 {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
2524sseq1i 3825 . . . . 5 ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2625reubii 3311 . . . 4 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2726biimpi 208 . . 3 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2822, 27syl6com 37 . 2 (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
293, 28simplbiim 500 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  ∃!wreu 3091  cdif 3766  wss 3769  {csn 4368  {cpr 4370  cfv 6101  Vtxcvtx 26231  Edgcedg 26282  USGraphcusgr 26385   FriendGraph cfrgr 27605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-frgr 27606
This theorem is referenced by:  frcond2  27616  frcond3  27618  4cyclusnfrgr  27641  frgrncvvdeqlem2  27649
  Copyright terms: Public domain W3C validator