Theorem frgr1v 30233

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 4466	. . . . 5 ⊢ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)
3		sneq 4589	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {𝑘} = {𝑁})
4	3	difeq2d 4079	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ({𝑁} ∖ {𝑁}))
5		difid 4329	. . . . . . . 8 ⊢ ({𝑁} ∖ {𝑁}) = ∅
6	4, 5	eqtrdi 2780	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ∅)
7		preq2 4688	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → {𝑥, 𝑘} = {𝑥, 𝑁})
8	7	preq1d 4693	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑁}, {𝑥, 𝑙}})
9	8	sseq1d 3969	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
10	9	reubidv 3363	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
11	6, 10	raleqbidv 3310	. . . . . 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 4629	. . . . 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 4671	. . . . 5 ⊢ (¬ 𝑁 ∈ V ↔ {𝑁} = ∅)
15		rzal 4462	. . . . 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 4072	. . . . . 6 ⊢ ((Vtx‘𝐺) = {𝑁} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝑁} ∖ {𝑘}))
20		reueq1 3379	. . . . . 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 3310	. . . . 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 3310	. . . 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 2729	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
26		eqid 2729	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
27	25, 26	isfrgr 30222	. 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 2109  ∀wral 3044  ∃!wreu 3343  Vcvv 3438   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  {csn 4579  {cpr 4581  ‘cfv 6486  Vtxcvtx 28959  Edgcedg 29010  USGraphcusgr 29112   FriendGraph cfrgr 30220

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-frgr 30221

This theorem is referenced by: (None)