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Theorem frcond1 27246
 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 27239 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4 preq2 4301 . . . . . . 7 (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
54preq1d 4306 . . . . . 6 (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
65sseq1d 3665 . . . . 5 (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
76reubidv 3156 . . . 4 (𝑘 = 𝐴 → (∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8 preq2 4301 . . . . . . 7 (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
98preq2d 4307 . . . . . 6 (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
109sseq1d 3665 . . . . 5 (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
1110reubidv 3156 . . . 4 (𝑙 = 𝐶 → (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12 simp1 1081 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐴𝑉)
13 sneq 4220 . . . . . 6 (𝑘 = 𝐴 → {𝑘} = {𝐴})
1413difeq2d 3761 . . . . 5 (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
1514adantl 481 . . . 4 (((𝐴𝑉𝐶𝑉𝐴𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16 necom 2876 . . . . . . . 8 (𝐴𝐶𝐶𝐴)
1716biimpi 206 . . . . . . 7 (𝐴𝐶𝐶𝐴)
1817anim2i 592 . . . . . 6 ((𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
19183adant1 1099 . . . . 5 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
20 eldifsn 4350 . . . . 5 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉𝐶𝐴))
2119, 20sylibr 224 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
227, 11, 12, 15, 21rspc2vd 27245 . . 3 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
23 prcom 4299 . . . . . . 7 {𝑏, 𝐴} = {𝐴, 𝑏}
2423preq1i 4303 . . . . . 6 {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
2524sseq1i 3662 . . . . 5 ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2625reubii 3158 . . . 4 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2726biimpi 206 . . 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 659 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃!wreu 2943   ∖ cdif 3604   ⊆ wss 3607  {csn 4210  {cpr 4212  ‘cfv 5926  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   FriendGraph cfrgr 27236 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-frgr 27237 This theorem is referenced by:  frcond2  27247  frcond3  27249  4cyclusnfrgr  27272  frgrncvvdeqlem2  27280
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