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Theorem frgrusgrfrcond 41433
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 41432 . . . 4 (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
4 simpl 471 . . . 4 ((𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸) → 𝐺 ∈ USGraph )
53, 4syl6bi 241 . . 3 (𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph ))
65pm2.43i 49 . 2 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
71, 2isfrgr 41432 . 2 (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
86, 4, 7pm5.21nii 366 1 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  ∃!wreu 2897  cdif 3536  wss 3539  {csn 4124  {cpr 4126  cfv 5790  Vtxcvtx 40231  Edgcedga 40353   USGraph cusgr 40381   FriendGraph cfrgr 41430
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-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-nul 4712
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-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-sbc 3402  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-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-frgr 41431
This theorem is referenced by:  frgrusgr  41434  frgr0v  41435  frgr0  41438  frcond1  41440  frcond3  41442  frgr1v  41443  nfrgr2v  41444  frgr3v  41447  2pthfrgrrn  41454  n4cyclfrgr  41463
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