Theorem clnbgr3stgrgrlic 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 the two graphs are locally isomorphic. (Contributed by AV, 29-Sep-2025.)

Hypotheses

Ref	Expression
clnbgr3stgrgrlic.n	⊢ 𝑁 ∈ ℕ0
clnbgr3stgrgrlic.v	⊢ 𝑉 = (Vtx‘𝐺)
clnbgr3stgrgrlic.w	⊢ 𝑊 = (Vtx‘𝐻)

Assertion

Ref	Expression
clnbgr3stgrgrlic	⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺 ≃𝑙𝑔𝑟 𝐻)

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

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem clnbgr3stgrgrlic

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgr3stgrgrlic.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6854	. . . . . . 7 ⊢ 𝑉 ∈ V
3		clnbgr3stgrgrlic.w	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘𝐻)
4	3	fvexi 6854	. . . . . . 7 ⊢ 𝑊 ∈ V
5	2, 4	pm3.2i 470	. . . . . 6 ⊢ (𝑉 ∈ V ∧ 𝑊 ∈ V)
6		breng 8904	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝑉 ≈ 𝑊 ↔ ∃𝑓 𝑓:𝑉–1-1-onto→𝑊))
7	5, 6	mp1i 13	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑉 ≈ 𝑊 ↔ ∃𝑓 𝑓:𝑉–1-1-onto→𝑊))
8		usgruhgr 29166	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 ∈ USGraph → 𝐻 ∈ UHGraph)
9	8	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → 𝐻 ∈ UHGraph)
10	9	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐻 ∈ UHGraph)
11	3	clnbgrssvtx 47825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 ClNeighbVtx (𝑓‘𝑥)) ⊆ 𝑊
12	11	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((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)
13	3	isubgruhgr 47861	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ∈ UHGraph ∧ (𝐻 ClNeighbVtx (𝑓‘𝑥)) ⊆ 𝑊) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ∈ UHGraph)
14	10, 12, 13	syl2an2r 685	. . . . . . . . . . . . . . . . 17 ⊢ ((((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)
15		f1of 6782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓:𝑉⟶𝑊)
16	15	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → 𝑓:𝑉⟶𝑊)
17		simp3 1138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
18	16, 17	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → (𝑓‘𝑥) ∈ 𝑊)
19		oveq2 7377	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ClNeighbVtx 𝑦) = (𝐻 ClNeighbVtx (𝑓‘𝑥)))
20	19	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))
21	20	breq1d 5112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) ↔ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
22	21	rspcv 3581	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑥) ∈ 𝑊 → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
23	18, 22	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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)))
24	23	3exp 1119	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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)))))
25	24	com34 91	. . . . . . . . . . . . . . . . . 18 ⊢ ((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)))))
26	25	3imp1 1348	. . . . . . . . . . . . . . . . 17 ⊢ ((((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))
27		gricsym 47914	. . . . . . . . . . . . . . . . 17 ⊢ ((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)))))
28	14, 26, 27	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ ((((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))))
29	28	anim1ci 616	. . . . . . . . . . . . . . 15 ⊢ (((((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)))))
30		grictr 47916	. . . . . . . . . . . . . . 15 ⊢ (((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))))
31	29, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((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))))
32	31	ex 412	. . . . . . . . . . . . 13 ⊢ ((((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)))))
33	32	ralimdva 3145	. . . . . . . . . . . 12 ⊢ (((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)))))
34	33	3exp 1119	. . . . . . . . . . 11 ⊢ ((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)))))))
35	34	com24 95	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (∀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 → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
36	35	imp32 418	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀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 → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
37	36	ancld 550	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀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 → (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
38	37	eximdv 1917	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁))) → (∃𝑓 𝑓:𝑉–1-1-onto→𝑊 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
39	38	ex 412	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → ((∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (∃𝑓 𝑓:𝑉–1-1-onto→𝑊 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
40	39	com23 86	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (∃𝑓 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(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
41	7, 40	sylbid 240	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑉 ≈ 𝑊 → ((∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
42	41	3impia 1117	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) → ((∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
43	42	3impib 1116	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
44	1, 3	dfgrlic2 47993	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
45	44	3adant3 1132	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
46	45	3ad2ant1 1133	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
47	43, 46	mpbird 257	1 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺 ≃𝑙𝑔𝑟 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911   class class class wbr 5102  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369   ≈ cen 8892  ℕ₀cn0 12418  Vtxcvtx 28976  UHGraphcuhgr 29036  USGraphcusgr 29129   ClNeighbVtx cclnbgr 47812   ISubGr cisubgr 47853   ≃_𝑔𝑟 cgric 47869  StarGrcstgr 47943   ≃_𝑙𝑔𝑟 cgrlic 47969

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-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-i2m1 11112  ax-1ne0 11113  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119

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-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-2 12225  df-vtx 28978  df-iedg 28979  df-uhgr 29038  df-upgr 29062  df-uspgr 29130  df-usgr 29131  df-clnbgr 47813  df-isubgr 47854  df-grim 47871  df-gric 47874  df-grlim 47970  df-grlic 47973

This theorem is referenced by:  gpg5grlic  48077