Theorem frgr1v 30252

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 482	. 2 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ USGraph)
2		ral0 4488	. . . . 5 ⊢ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)
3		sneq 4611	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {𝑘} = {𝑁})
4	3	difeq2d 4101	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ({𝑁} ∖ {𝑁}))
5		difid 4351	. . . . . . . 8 ⊢ ({𝑁} ∖ {𝑁}) = ∅
6	4, 5	eqtrdi 2786	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ∅)
7		preq2 4710	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → {𝑥, 𝑘} = {𝑥, 𝑁})
8	7	preq1d 4715	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑁}, {𝑥, 𝑙}})
9	8	sseq1d 3990	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
10	9	reubidv 3377	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
11	6, 10	raleqbidv 3325	. . . . . 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 4651	. . . . 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 258	. . . 4 ⊢ (𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
14		snprc 4693	. . . . 5 ⊢ (¬ 𝑁 ∈ V ↔ {𝑁} = ∅)
15		rzal 4484	. . . . 5 ⊢ ({𝑁} = ∅ → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
16	14, 15	sylbi 217	. . . 4 ⊢ (¬ 𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
17	13, 16	pm2.61i 182	. . 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 4094	. . . . . 6 ⊢ ((Vtx‘𝐺) = {𝑁} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝑁} ∖ {𝑘}))
20		reueq1 3396	. . . . . 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 3325	. . . . 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 3325	. . . 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 481	. . 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 258	. 2 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
25		eqid 2735	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
26		eqid 2735	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
27	25, 26	isfrgr 30241	. 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 583	1 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃!wreu 3357  Vcvv 3459   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  {csn 4601  {cpr 4603  ‘cfv 6531  Vtxcvtx 28975  Edgcedg 29026  USGraphcusgr 29128   FriendGraph cfrgr 30239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-frgr 30240

This theorem is referenced by: (None)