Theorem clnbgr3stgrgrlim 48004

Description: If all (closed) neighborhoods of the vertices in two simple graphs with the same order induce a subgraph which is isomorphic to an 𝑁-star, then any bijection between the vertices is a local isomorphism between the two graphs. (Contributed by AV, 28-Dec-2025.)

Hypotheses

Ref	Expression
clnbgr3stgrgrlim.n	⊢ 𝑁 ∈ ℕ0
clnbgr3stgrgrlim.v	⊢ 𝑉 = (Vtx‘𝐺)
clnbgr3stgrgrlim.w	⊢ 𝑊 = (Vtx‘𝐻)

Assertion

Ref	Expression
clnbgr3stgrgrlim	⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺   𝑥,𝐻,𝑦   𝑥,𝑁,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦

Allowed substitution hints:   𝐺(𝑦)   𝑉(𝑦)

Proof of Theorem clnbgr3stgrgrlim

Step	Hyp	Ref	Expression
1		simp13 1206	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐹:𝑉–1-1-onto→𝑊)
2		usgruhgr 29149	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ USGraph → 𝐻 ∈ UHGraph)
3	2	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐻 ∈ UHGraph)
4	3	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐻 ∈ UHGraph)
5		clnbgr3stgrgrlim.w	. . . . . . . . . . . . 13 ⊢ 𝑊 = (Vtx‘𝐻)
6	5	clnbgrssvtx 47816	. . . . . . . . . . . 12 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝑥)) ⊆ 𝑊
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (𝐻 ClNeighbVtx (𝐹‘𝑥)) ⊆ 𝑊)
8	5	isubgruhgr 47853	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ UHGraph ∧ (𝐻 ClNeighbVtx (𝐹‘𝑥)) ⊆ 𝑊) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ∈ UHGraph)
9	4, 7, 8	syl2an2r 685	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ∈ UHGraph)
10		f1of 6768	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉⟶𝑊)
11	10	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐹:𝑉⟶𝑊)
12	11	ffvelcdmda 7022	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ 𝑊)
13		oveq2 7361	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐻 ClNeighbVtx 𝑦) = (𝐻 ClNeighbVtx (𝐹‘𝑥)))
14	13	oveq2d 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))
15	14	breq1d 5105	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) ↔ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
16	15	rspcv 3575	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ 𝑊 → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
17	12, 16	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
18	17	impancom 451	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝑥 ∈ 𝑉 → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
19	18	imp 406	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁))
20		gricsym 47906	. . . . . . . . . 10 ⊢ ((𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ∈ UHGraph → ((𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁) → (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥)))))
21	9, 19, 20	sylc 65	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))
22	21	anim1ci 616	. . . . . . . 8 ⊢ (((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) ∧ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁)) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥)))))
23		grictr 47908	. . . . . . . 8 ⊢ (((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥)))) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))
24	22, 23	syl 17	. . . . . . 7 ⊢ (((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) ∧ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))
25	24	ex 412	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥)))))
26	25	ralimdva 3141	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥)))))
27	26	ex 412	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))))
28	27	com23 86	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) → (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))))
29	28	3imp 1110	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))
30		clnbgr3stgrgrlim.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
31	30	fvexi 6840	. . . . . . 7 ⊢ 𝑉 ∈ V
32	31	a1i 11	. . . . . 6 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝑉 ∈ V)
33	10, 32	fexd 7167	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 ∈ V)
34	33	3anim3i 1154	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) → (𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹 ∈ V))
35	34	3ad2ant1 1133	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹 ∈ V))
36	30, 5	isgrlim 47967	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹 ∈ V) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))))
37	35, 36	syl 17	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑥))))))
38	1, 29, 37	mpbir2and 713	1 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438   ⊆ wss 3905   class class class wbr 5095  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  ℕ₀cn0 12402  Vtxcvtx 28959  UHGraphcuhgr 29019  USGraphcusgr 29112   ClNeighbVtx cclnbgr 47803   ISubGr cisubgr 47845   ≃_𝑔𝑟 cgric 47861  StarGrcstgr 47936   GraphLocIso cgrlim 47961

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-i2m1 11096  ax-1ne0 11097  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-po 5531  df-so 5532  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-1o 8395  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-2 12209  df-vtx 28961  df-iedg 28962  df-uhgr 29021  df-upgr 29045  df-uspgr 29113  df-usgr 29114  df-clnbgr 47804  df-isubgr 47846  df-grim 47863  df-gric 47866  df-grlim 47963

This theorem is referenced by:  gpg5grlim  48078