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Theorem isfrgr 30288
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 6919 . . 3 (Vtx‘𝑔) ∈ V
2 fvex 6919 . . 3 (Edg‘𝑔) ∈ V
3 fveq2 6906 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
43eqeq2d 2745 . . . . . 6 (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = (Vtx‘𝐺)))
5 isfrgr.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
65eqcomi 2743 . . . . . . 7 (Vtx‘𝐺) = 𝑉
76eqeq2i 2747 . . . . . 6 (𝑣 = (Vtx‘𝐺) ↔ 𝑣 = 𝑉)
84, 7bitrdi 287 . . . . 5 (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = 𝑉))
9 fveq2 6906 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
109eqeq2d 2745 . . . . . 6 (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = (Edg‘𝐺)))
11 isfrgr.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
1211eqcomi 2743 . . . . . . 7 (Edg‘𝐺) = 𝐸
1312eqeq2i 2747 . . . . . 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 482 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
17 difeq1 4128 . . . . . . 7 (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
1817adantr 480 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
19 reueq1 3414 . . . . . . . 8 (𝑣 = 𝑉 → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
2019adantr 480 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
21 sseq2 4021 . . . . . . . . 9 (𝑒 = 𝐸 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2221adantl 481 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2322reubidv 3395 . . . . . . 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 3343 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2616, 25raleqbidv 3343 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
2715, 26biimtrdi 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 3875 . 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 30287 . 2 FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
3028, 29elrab2 3697 1 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  ∃!wreu 3375  [wsbc 3790  cdif 3959  wss 3962  {csn 4630  {cpr 4632  cfv 6562  Vtxcvtx 29027  Edgcedg 29078  USGraphcusgr 29180   FriendGraph cfrgr 30286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-frgr 30287
This theorem is referenced by:  frgrusgr  30289  frgr0v  30290  frgr0  30293  frcond1  30294  frgr1v  30299  nfrgr2v  30300  frgr3v  30303  2pthfrgrrn  30310  n4cyclfrgr  30319
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