Theorem isfrgr 30854

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 6896	. . 3 ⊢ (Vtx‘𝑔) ∈ V
2		fvex 6896	. . 3 ⊢ (Edg‘𝑔) ∈ V
3		fveq2 6883	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4	3	eqeq2d 2772	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = (Vtx‘𝐺)))
5		isfrgr.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	eqcomi 2770	. . . . . . 7 ⊢ (Vtx‘𝐺) = 𝑉
7	6	eqeq2i 2774	. . . . . 6 ⊢ (𝑣 = (Vtx‘𝐺) ↔ 𝑣 = 𝑉)
8	4, 7	bitrdi 290	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑣 = (Vtx‘𝑔) ↔ 𝑣 = 𝑉))
9		fveq2 6883	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
10	9	eqeq2d 2772	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = (Edg‘𝐺)))
11		isfrgr.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
12	11	eqcomi 2770	. . . . . . 7 ⊢ (Edg‘𝐺) = 𝐸
13	12	eqeq2i 2774	. . . . . 6 ⊢ (𝑒 = (Edg‘𝐺) ↔ 𝑒 = 𝐸)
14	10, 13	bitrdi 290	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑒 = (Edg‘𝑔) ↔ 𝑒 = 𝐸))
15	8, 14	anbi12d 644	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑣 = (Vtx‘𝑔) ∧ 𝑒 = (Edg‘𝑔)) ↔ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)))
16		simpl 488	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
17		difeq1 4067	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
18	17	adantr 486	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
19		reueq1 3398	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
20	19	adantr 486	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ ∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒))
21		sseq2 3957	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
22	21	adantl 487	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
23	22	reubidv 3382	. . . . . . 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 3335	. . . . 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 3335	. . . 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	biimtrdi 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 3815	. 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 30853	. 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 3649	1 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃!wreu 3364  [wsbc 3739   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  {cpr 4586  ‘cfv 6537  Vtxcvtx 29567  Edgcedg 29618  USGraphcusgr 29723   FriendGraph cfrgr 30852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6493  df-fv 6545  df-frgr 30853

This theorem is used by:  frgrusgr  30855  frgr0v  30856  frgr0  30859  frcond1  30860  frgr1v  30865  nfrgr2v  30866  frgr3v  30869  2pthfrgrrn  30876  n4cyclfrgr  30885