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Theorem frisusgrapr 26311
Description: A friendship graph is an undirected simple graph without loops with special properties. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frisusgrapr (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
Distinct variable groups:   𝑘,𝑙,𝑥,𝑉   𝑘,𝐸,𝑙,𝑥

Proof of Theorem frisusgrapr
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-frgra 26309 . . . . 5 FriendGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒)}
21relopabi 5155 . . . 4 Rel FriendGrph
32brrelexi 5071 . . 3 (𝑉 FriendGrph 𝐸𝑉 ∈ V)
42brrelex2i 5072 . . 3 (𝑉 FriendGrph 𝐸𝐸 ∈ V)
5 isfrgra 26310 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 FriendGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
63, 4, 5syl2anc 690 . 2 (𝑉 FriendGrph 𝐸 → (𝑉 FriendGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
76ibi 254 1 (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  wral 2895  ∃!wreu 2897  Vcvv 3172  cdif 3536  wss 3539  {csn 4124  {cpr 4126   class class class wbr 4577  ran crn 5028   USGrph cusg 25652   FriendGrph cfrgra 26308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-frgra 26309
This theorem is referenced by:  frisusgra  26312  frgraunss  26315  frisusgranb  26317  2pthfrgrarn  26329  n4cyclfrgra  26338
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