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Theorem isfrgr 28036
 Description: The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.)
Hypotheses
Ref Expression
isfrgr.v 𝑉 = (Vtx‘𝐺)
isfrgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
isfrgr (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
Distinct variable groups:   𝑘,𝐸,𝑙,𝑥   𝑘,𝑉,𝑙,𝑥
Allowed substitution hints:   𝐺(𝑥,𝑘,𝑙)

Proof of Theorem isfrgr
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6666 . . 3 (Vtx‘𝑔) ∈ V
2 fvex 6666 . . 3 (Edg‘𝑔) ∈ V
3 fveq2 6653 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43eqeq2d 2835 . . . . . 6 (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = (Vtx‘𝐺)))
5 isfrgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
65eqcomi 2833 . . . . . . 7 (Vtx‘𝐺) = 𝑉
76eqeq2i 2837 . . . . . 6 (𝑣 = (Vtx‘𝐺) ↔ 𝑣 = 𝑉)
84, 7syl6bb 290 . . . . 5 (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = 𝑉))
9 fveq2 6653 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
109eqeq2d 2835 . . . . . 6 (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = (Edg‘𝐺)))
11 isfrgr.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
1211eqcomi 2833 . . . . . . 7 (Edg‘𝐺) = 𝐸
1312eqeq2i 2837 . . . . . 6 (𝑒 = (Edg‘𝐺) ↔ 𝑒 = 𝐸)
1410, 13syl6bb 290 . . . . 5 (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = 𝐸))
158, 14anbi12d 633 . . . 4 (𝑔 = 𝐺 → ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (Edg‘𝑔)) ↔ (𝑣 = 𝑉𝑒 = 𝐸)))
16 simpl 486 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
17 difeq1 4076 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
1817adantr 484 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
19 reueq1 3398 . . . . . . . 8 (𝑣 = 𝑉 → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
2019adantr 484 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
21 sseq2 3977 . . . . . . . . 9 (𝑒 = 𝐸 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2221adantl 485 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2322reubidv 3380 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2420, 23bitrd 282 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2518, 24raleqbidv 3392 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2616, 25raleqbidv 3392 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2715, 26syl6bi 256 . . 3 (𝑔 = 𝐺 → ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (Edg‘𝑔)) → (∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
281, 2, 27sbc2iedv 3834 . 2 (𝑔 = 𝐺 → ([(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
29 df-frgr 28035 . 2 FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
3028, 29elrab2 3668 1 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3132  ∃!wreu 3134  [wsbc 3757   ∖ cdif 3915   ⊆ wss 3918  {csn 4548  {cpr 4550  ‘cfv 6338  Vtxcvtx 26780  Edgcedg 26831  USGraphcusgr 26933   FriendGraph cfrgr 28034 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5193 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-iota 6297  df-fv 6346  df-frgr 28035 This theorem is referenced by:  frgrusgr  28037  frgr0v  28038  frgr0  28041  frcond1  28042  frgr1v  28047  nfrgr2v  28048  frgr3v  28051  2pthfrgrrn  28058  n4cyclfrgr  28067
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