Theorem frgrusgrfrcond 27809

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 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{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

Step	Hyp	Ref	Expression
1		isfrgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		isfrgr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	isfrgr 27808	. . . 4 ⊢ (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
4		simpl 475	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸) → 𝐺 ∈ USGraph)
5	3, 4	syl6bi 245	. . 3 ⊢ (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph))
6	5	pm2.43i 52	. 2 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
7	1, 2	isfrgr 27808	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
8	6, 4, 7	pm5.21nii 371	1 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3083  ∃!wreu 3085   ∖ cdif 3821   ⊆ wss 3824  {csn 4436  {cpr 4438  ‘cfv 6186  Vtxcvtx 26500  Edgcedg 26551  USGraphcusgr 26653   FriendGraph cfrgr 27806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745  ax-nul 5064

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-iota 6150  df-fv 6194  df-frgr 27807

This theorem is referenced by:  frgrusgr  27810  frgr0v  27811  frgr0  27814  frcond1  27816  frgr1v  27821  nfrgr2v  27822  frgr3v  27825  2pthfrgrrn  27832  n4cyclfrgr  27841