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Theorem frgrusgrfrcond 27239
Description: A friendship graph is a simple graph which fulfils the friendship condition. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
isfrgr.v 𝑉 = (Vtx‘𝐺)
isfrgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrusgrfrcond (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
Distinct variable groups:   𝑘,𝑙,𝑥,𝐺   𝑘,𝑉,𝑙,𝑥
Allowed substitution hints:   𝐸(𝑥,𝑘,𝑙)

Proof of Theorem frgrusgrfrcond
StepHypRef Expression
1 isfrgr.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 isfrgr.e . . . . 5 𝐸 = (Edg‘𝐺)
31, 2isfrgr 27238 . . . 4 (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
4 simpl 472 . . . 4 ((𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸) → 𝐺 ∈ USGraph)
53, 4syl6bi 243 . . 3 (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph))
65pm2.43i 52 . 2 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
71, 2isfrgr 27238 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
86, 4, 7pm5.21nii 367 1 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  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-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  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:  frgrusgr  27240  frgr0v  27241  frgr0  27244  frcond1  27246  frgr1v  27251  nfrgr2v  27252  frgr3v  27255  2pthfrgrrn  27262  n4cyclfrgr  27271
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