Theorem isfrgr 28045

Description: The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
isfrgr	⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

Distinct variable groups:   𝑘,𝐸,𝑙,𝑥   𝑘,𝑉,𝑙,𝑥

Allowed substitution hints:   𝐺(𝑥,𝑘,𝑙)

Proof of Theorem isfrgr

Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6658	. . 3 ⊢ (Vtx‘𝑔) ∈ V
2		fvex 6658	. . 3 ⊢ (Edg‘𝑔) ∈ V
3		fveq2 6645	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	eqeq2d 2809	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = (Vtx‘𝐺)))
5		isfrgr.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	eqcomi 2807	. . . . . . 7 ⊢ (Vtx‘𝐺) = 𝑉
7	6	eqeq2i 2811	. . . . . 6 ⊢ (𝑣 = (Vtx‘𝐺) ↔ 𝑣 = 𝑉)
8	4, 7	syl6bb 290	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = 𝑉))
9		fveq2 6645	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
10	9	eqeq2d 2809	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = (Edg‘𝐺)))
11		isfrgr.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
12	11	eqcomi 2807	. . . . . . 7 ⊢ (Edg‘𝐺) = 𝐸
13	12	eqeq2i 2811	. . . . . 6 ⊢ (𝑒 = (Edg‘𝐺) ↔ 𝑒 = 𝐸)
14	10, 13	syl6bb 290	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = 𝐸))
15	8, 14	anbi12d 633	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (Edg‘𝑔)) ↔ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)))
16		simpl 486	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
17		difeq1 4043	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
18	17	adantr 484	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
19		reueq1 3360	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
20	19	adantr 484	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
21		sseq2 3941	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
22	21	adantl 485	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
23	22	reubidv 3342	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
24	20, 23	bitrd 282	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
25	18, 24	raleqbidv 3354	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
26	16, 25	raleqbidv 3354	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
27	15, 26	syl6bi 256	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (Edg‘𝑔)) → (∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)))
28	1, 2, 27	sbc2iedv 3797	. 2 ⊢ (𝑔 = 𝐺 → ([(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
29		df-frgr 28044	. 2 ⊢ FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
30	28, 29	elrab2 3631	1 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃!wreu 3108  [wsbc 3720   ∖ cdif 3878   ⊆ wss 3881  {csn 4525  {cpr 4527  ‘cfv 6324  Vtxcvtx 26789  Edgcedg 26840  USGraphcusgr 26942   FriendGraph cfrgr 28043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-frgr 28044

This theorem is referenced by:  frgrusgr  28046  frgr0v  28047  frgr0  28050  frcond1  28051  frgr1v  28056  nfrgr2v  28057  frgr3v  28060  2pthfrgrrn  28067  n4cyclfrgr  28076