Theorem frgr1v 30659

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.

Step	Hyp	Ref	Expression
1		simpl 488	. 2 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ USGraph)
2		ral0 4464	. . . . 5 ⊢ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)
3		sneq 4604	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {𝑘} = {𝑁})
4	3	difeq2d 4084	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ({𝑁} ∖ {𝑁}))
5		difid 4335	. . . . . . . 8 ⊢ ({𝑁} ∖ {𝑁}) = ∅
6	4, 5	eqtrdi 2817	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ∅)
7		preq2 4705	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → {𝑥, 𝑘} = {𝑥, 𝑁})
8	7	preq1d 4710	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑁}, {𝑥, 𝑙}})
9	8	sseq1d 3971	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
10	9	reubidv 3388	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
11	6, 10	raleqbidv 3341	. . . . . 6 ⊢ (𝑘 = 𝑁 → (∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
12	11	ralsng 4646	. . . . 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)))
13	2, 12	mpbiri 261	. . . 4 ⊢ (𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
14		snprc 4688	. . . . 5 ⊢ (¬ 𝑁 ∈ V ↔ {𝑁} = ∅)
15		rzal 4460	. . . . 5 ⊢ ({𝑁} = ∅ → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
16	14, 15	sylbi 220	. . . 4 ⊢ (¬ 𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
17	13, 16	pm2.61i 184	. . 3 ⊢ ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)
18		id 23	. . . . 5 ⊢ ((Vtx‘𝐺) = {𝑁} → (Vtx‘𝐺) = {𝑁})
19		difeq1 4077	. . . . . 6 ⊢ ((Vtx‘𝐺) = {𝑁} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝑁} ∖ {𝑘}))
20		reueq1 3404	. . . . . 6 ⊢ ((Vtx‘𝐺) = {𝑁} → (∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
21	19, 20	raleqbidv 3341	. . . . 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)))
22	18, 21	raleqbidv 3341	. . . 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)))
23	22	adantl 487	. . 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)))
24	17, 23	mpbiri 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 2766	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
26		eqid 2766	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
27	25, 26	isfrgr 30648	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
28	1, 24, 27	sylanbrc 595	1 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃!wreu 3370  Vcvv 3458   ∖ cdif 3905   ⊆ wss 3908  ∅c0 4289  {csn 4594  {cpr 4596  ‘cfv 6543  Vtxcvtx 29383  Edgcedg 29434  USGraphcusgr 29536   FriendGraph cfrgr 30646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-frgr 30647

This theorem is used by: (None)