Theorem frisusgrapr 26518

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 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{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.

Step	Hyp	Ref	Expression
1		df-frgra 26516	. . . . 5 ⊢ FriendGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒)}
2	1	relopabi 5167	. . . 4 ⊢ Rel FriendGrph
3	2	brrelexi 5082	. . 3 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 ∈ V)
4	2	brrelex2i 5083	. . 3 ⊢ (𝑉 FriendGrph 𝐸 → 𝐸 ∈ V)
5		isfrgra 26517	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 FriendGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
6	3, 4, 5	syl2anc 691	. 2 ⊢ (𝑉 FriendGrph 𝐸 → (𝑉 FriendGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
7	6	ibi 255	1 ⊢ (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   FriendGrph cfrgra 26515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-frgra 26516

This theorem is referenced by:  frisusgra  26519  frgraunss  26522  frisusgranb  26524  2pthfrgrarn  26536  n4cyclfrgra  26545