Theorem frcond1 30246

Description: The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
frcond1.v	⊢ 𝑉 = (Vtx‘𝐺)
frcond1.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frcond1	⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝐺,𝑏   𝑉,𝑏

Proof of Theorem frcond1

Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frcond1.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		frcond1.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	isfrgr 30240	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4		preq2 4684	. . . . . . 7 ⊢ (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
5	4	preq1d 4689	. . . . . 6 ⊢ (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
6	5	sseq1d 3961	. . . . 5 ⊢ (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
7	6	reubidv 3362	. . . 4 ⊢ (𝑘 = 𝐴 → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8		preq2 4684	. . . . . . 7 ⊢ (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
9	8	preq2d 4690	. . . . . 6 ⊢ (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
10	9	sseq1d 3961	. . . . 5 ⊢ (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
11	10	reubidv 3362	. . . 4 ⊢ (𝑙 = 𝐶 → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12		simp1 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ 𝑉)
13		sneq 4583	. . . . . 6 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
14	13	difeq2d 4073	. . . . 5 ⊢ (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
15	14	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16		necom 2981	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
17	16	biimpi 216	. . . . . . 7 ⊢ (𝐴 ≠ 𝐶 → 𝐶 ≠ 𝐴)
18	17	anim2i 617	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
19	18	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
20		eldifsn 4735	. . . . 5 ⊢ (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
21	19, 20	sylibr 234	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
22	7, 11, 12, 15, 21	rspc2vd 3893	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
23		prcom 4682	. . . . . . 7 ⊢ {𝑏, 𝐴} = {𝐴, 𝑏}
24	23	preq1i 4686	. . . . . 6 ⊢ {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
25	24	sseq1i 3958	. . . . 5 ⊢ ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
26	25	reubii 3355	. . . 4 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
27	26	biimpi 216	. . 3 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
28	22, 27	syl6com 37	. 2 ⊢ (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
29	3, 28	simplbiim 504	1 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃!wreu 3344   ∖ cdif 3894   ⊆ wss 3897  {csn 4573  {cpr 4575  ‘cfv 6481  Vtxcvtx 28974  Edgcedg 29025  USGraphcusgr 29127   FriendGraph cfrgr 30238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-frgr 30239

This theorem is referenced by:  frcond2  30247  frcond3  30249  4cyclusnfrgr  30272  frgrncvvdeqlem2  30280