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Theorem frgr1v 28386
Description: Any graph with (at most) one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
Assertion
Ref Expression
frgr1v ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph )

Proof of Theorem frgr1v
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 486 . 2 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ USGraph)
2 ral0 4441 . . . . 5 𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)
3 sneq 4568 . . . . . . . . 9 (𝑘 = 𝑁 → {𝑘} = {𝑁})
43difeq2d 4054 . . . . . . . 8 (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ({𝑁} ∖ {𝑁}))
5 difid 4302 . . . . . . . 8 ({𝑁} ∖ {𝑁}) = ∅
64, 5eqtrdi 2796 . . . . . . 7 (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ∅)
7 preq2 4667 . . . . . . . . . 10 (𝑘 = 𝑁 → {𝑥, 𝑘} = {𝑥, 𝑁})
87preq1d 4672 . . . . . . . . 9 (𝑘 = 𝑁 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑁}, {𝑥, 𝑙}})
98sseq1d 3949 . . . . . . . 8 (𝑘 = 𝑁 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
109reubidv 3316 . . . . . . 7 (𝑘 = 𝑁 → (∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
116, 10raleqbidv 3328 . . . . . 6 (𝑘 = 𝑁 → (∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
1211ralsng 4606 . . . . 5 (𝑁 ∈ V → (∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
132, 12mpbiri 261 . . . 4 (𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
14 snprc 4650 . . . . 5 𝑁 ∈ V ↔ {𝑁} = ∅)
15 rzal 4437 . . . . 5 ({𝑁} = ∅ → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
1614, 15sylbi 220 . . . 4 𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
1713, 16pm2.61i 185 . . 3 𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)
18 id 22 . . . . 5 ((Vtx‘𝐺) = {𝑁} → (Vtx‘𝐺) = {𝑁})
19 difeq1 4047 . . . . . 6 ((Vtx‘𝐺) = {𝑁} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝑁} ∖ {𝑘}))
20 reueq1 3336 . . . . . 6 ((Vtx‘𝐺) = {𝑁} → (∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
2119, 20raleqbidv 3328 . . . . 5 ((Vtx‘𝐺) = {𝑁} → (∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
2218, 21raleqbidv 3328 . . . 4 ((Vtx‘𝐺) = {𝑁} → (∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
2322adantl 485 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → (∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
2417, 23mpbiri 261 . 2 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
25 eqid 2739 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
26 eqid 2739 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
2725, 26isfrgr 28375 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
281, 24, 27sylanbrc 586 1 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  wral 3064  ∃!wreu 3066  Vcvv 3423  cdif 3880  wss 3883  c0 4254  {csn 4558  {cpr 4560  cfv 6401  Vtxcvtx 27119  Edgcedg 27170  USGraphcusgr 27272   FriendGraph cfrgr 28373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-nul 5216
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6359  df-fv 6409  df-frgr 28374
This theorem is referenced by: (None)
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