Theorem clnbgr3stgrgrlic 47932

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 6887	. . . . . . 7 ⊢ 𝑉 ∈ V
3		clnbgr3stgrgrlic.w	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘𝐻)
4	3	fvexi 6887	. . . . . . 7 ⊢ 𝑊 ∈ V
5	2, 4	pm3.2i 470	. . . . . 6 ⊢ (𝑉 ∈ V ∧ 𝑊 ∈ V)
6		breng 8963	. . . . . 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 29099	. . . . . . . . . . . . . . . . . . . 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 47771	. . . . . . . . . . . . . . . . . . 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 47807	. . . . . . . . . . . . . . . . . 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 6815	. . . . . . . . . . . . . . . . . . . . . . 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 7072	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → (𝑓‘𝑥) ∈ 𝑊)
19		oveq2 7408	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ClNeighbVtx 𝑦) = (𝐻 ClNeighbVtx (𝑓‘𝑥)))
20	19	oveq2d 7416	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))
21	20	breq1d 5127	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) ↔ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
22	21	rspcv 3595	. . . . . . . . . . . . . . . . . . . . 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 1347	. . . . . . . . . . . . . . . . 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 47843	. . . . . . . . . . . . . . . . 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 47845	. . . . . . . . . . . . . . 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 3150	. . . . . . . . . . . 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 1916	. . . . . . 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 47921	. . . 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 1539  ∃wex 1778   ∈ wcel 2107  ∀wral 3050  Vcvv 3457   ⊆ wss 3924   class class class wbr 5117  ⟶wf 6524  –1-1-onto→wf1o 6527  ‘cfv 6528  (class class class)co 7400   ≈ cen 8951  ℕ₀cn0 12494  Vtxcvtx 28909  UHGraphcuhgr 28969  USGraphcusgr 29062   ClNeighbVtx cclnbgr 47758   ISubGr cisubgr 47799   ≃_𝑔𝑟 cgric 47815  StarGrcstgr 47871   ≃_𝑙𝑔𝑟 cgrlic 47897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724  ax-resscn 11179  ax-1cn 11180  ax-icn 11181  ax-addcl 11182  ax-addrcl 11183  ax-mulcl 11184  ax-mulrcl 11185  ax-i2m1 11190  ax-1ne0 11191  ax-rrecex 11194  ax-cnre 11195  ax-pre-lttri 11196  ax-pre-lttrn 11197

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-po 5559  df-so 5560  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-suc 6356  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-1o 8475  df-er 8714  df-map 8837  df-en 8955  df-dom 8956  df-sdom 8957  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-2 12296  df-vtx 28911  df-iedg 28912  df-uhgr 28971  df-upgr 28995  df-uspgr 29063  df-usgr 29064  df-clnbgr 47759  df-isubgr 47800  df-grim 47817  df-gric 47820  df-grlim 47898  df-grlic 47901

This theorem is referenced by:  gpg5grlic  48000