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Theorem isfrgr 29207
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 6856 . . 3 (Vtx‘𝑔) ∈ V
2 fvex 6856 . . 3 (Edg‘𝑔) ∈ V
3 fveq2 6843 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43eqeq2d 2748 . . . . . 6 (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = (Vtx‘𝐺)))
5 isfrgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
65eqcomi 2746 . . . . . . 7 (Vtx‘𝐺) = 𝑉
76eqeq2i 2750 . . . . . 6 (𝑣 = (Vtx‘𝐺) ↔ 𝑣 = 𝑉)
84, 7bitrdi 287 . . . . 5 (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = 𝑉))
9 fveq2 6843 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
109eqeq2d 2748 . . . . . 6 (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = (Edg‘𝐺)))
11 isfrgr.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
1211eqcomi 2746 . . . . . . 7 (Edg‘𝐺) = 𝐸
1312eqeq2i 2750 . . . . . 6 (𝑒 = (Edg‘𝐺) ↔ 𝑒 = 𝐸)
1410, 13bitrdi 287 . . . . 5 (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = 𝐸))
158, 14anbi12d 632 . . . 4 (𝑔 = 𝐺 → ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (Edg‘𝑔)) ↔ (𝑣 = 𝑉𝑒 = 𝐸)))
16 simpl 484 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
17 difeq1 4076 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
1817adantr 482 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
19 reueq1 3393 . . . . . . . 8 (𝑣 = 𝑉 → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
2019adantr 482 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
21 sseq2 3971 . . . . . . . . 9 (𝑒 = 𝐸 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2221adantl 483 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2322reubidv 3372 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2420, 23bitrd 279 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2518, 24raleqbidv 3320 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2616, 25raleqbidv 3320 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2715, 26syl6bi 253 . . 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 3825 . 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 29206 . 2 FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
3028, 29elrab2 3649 1 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  ∃!wreu 3352  [wsbc 3740  cdif 3908  wss 3911  {csn 4587  {cpr 4589  cfv 6497  Vtxcvtx 27950  Edgcedg 28001  USGraphcusgr 28103   FriendGraph cfrgr 29205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-frgr 29206
This theorem is referenced by:  frgrusgr  29208  frgr0v  29209  frgr0  29212  frcond1  29213  frgr1v  29218  nfrgr2v  29219  frgr3v  29222  2pthfrgrrn  29229  n4cyclfrgr  29238
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