Theorem frgr1v 27135

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 473	. 2 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ USGraph )
2		ral0 4076	. . . . 5 ⊢ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)
3		sneq 4187	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {𝑘} = {𝑁})
4	3	difeq2d 3728	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ({𝑁} ∖ {𝑁}))
5		difid 3948	. . . . . . . 8 ⊢ ({𝑁} ∖ {𝑁}) = ∅
6	4, 5	syl6eq 2672	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({𝑁} ∖ {𝑘}) = ∅)
7		preq2 4269	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → {𝑥, 𝑘} = {𝑥, 𝑁})
8	7	preq1d 4274	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑁}, {𝑥, 𝑙}})
9	8	sseq1d 3632	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
10	9	reubidv 3126	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝑁} {{𝑥, 𝑁}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
11	6, 10	raleqbidv 3152	. . . . . 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 4218	. . . . 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 248	. . . 4 ⊢ (𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
14		snprc 4253	. . . . 5 ⊢ (¬ 𝑁 ∈ V ↔ {𝑁} = ∅)
15		rzal 4073	. . . . 5 ⊢ ({𝑁} = ∅ → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
16	14, 15	sylbi 207	. . . 4 ⊢ (¬ 𝑁 ∈ V → ∀𝑘 ∈ {𝑁}∀𝑙 ∈ ({𝑁} ∖ {𝑘})∃!𝑥 ∈ {𝑁} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
17	13, 16	pm2.61i 176	. . 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 3721	. . . . . 6 ⊢ ((Vtx‘𝐺) = {𝑁} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝑁} ∖ {𝑘}))
20		reueq1 3140	. . . . . 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 3152	. . . . 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 3152	. . . 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 482	. . 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 248	. 2 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
25		eqid 2622	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
26		eqid 2622	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
27	25, 26	frgrusgrfrcond 27123	. 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 698	1 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃!wreu 2914  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ‘cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-frgr 27121

This theorem is referenced by: (None)